


mKBk 






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LIBRARY OF CONGRESS. 
®^A5%jri$t T$a 



Shelf^YO. 






UNITED STATES OF AMERICA. 



KEY 



WELLS' ESSENTIALS OF TKIGONOMETKY. 



/ 



WEBSTER WELLS, S.B., 

ASSOCIATE PROFESSOR OF MATHEMATICS IN THE MASSACHUSETTS 
INSTITUTE OF TECHNOLOGY. 






r 





LEACH, SHEWELL, & SANBORN. 

BOSTON AND NEW YORK. 



v< 



Copyright, 1888, 
By WEBSTER WELLS. 



Typography by Presswork by 

J. S. Cushing & Co. Berwick & Smith. 



KEY 



WELLS' ESSENTIALS OF TRIGONOMETRY. 





CHAPTER I. 






Art. 9. — Page 3. 




1. 


135° = ^. 5. 29° 15' 
4 


.13* 
" 80* 


2. 


198 o = lljr. 6# 1740 22' 30" = 
10 


31*- 
" 32 


3. 


11° 15' = — • 7. 128° 34V = 
16 T 


.5tt 

7 * 


4. 


37° 30' = §JL " 8. 92° 48' 45" = 
24 

9. - = - X 57.2957795° ... = 28.6478897° ... 
2 2 

= 28° 38.873382' ... =28° 38' 52.40292" - 

10. ^ = § of 180° = 108°. 
5 5 

n 37tt = 37 f 18QO = 2220t 
30 30 

12. lLZ = *> of 180° = 225°. 


33 7T 

:_ 6T' 



4 4 

13. - = - X 57.2957795° ... = 42.9718346° ... 
4 4 

= 42° 58.310076' ... =42° 58' 18.60456" ... 

14. 2 = 2 X 57.2957795° ... = 114.591559° ... 

= 114° 35.49354' -".. = 114° 35' 29.6124" ... 



2 KEY TO ESSENTIALS OF TRIGONOMETRY. 

15. 2 7 r-l = 27r_l > ?-? = ? of 180° = 120°. 

3 3 3 3 3 

i= - X 57.2957795° ... = 19.0985932° »• 
o o 

= 19° 5.915592' ... = 19° 5' 54.93552" ... 
2tt-1 



3 



= 120° - 19° 5' 54.93552" ... = 100° 54' 5.06448" ... 



16. 7r "- 1 = ---. - = i of 180° = 45°. 
4 4 4 4 4 

- = - X 57.2957795° ... = 14.3239449° ... 
4 4 

= 14° 19.436694' ••• =14° 19' 26.20164" • • • 

. 7T-1 



: 45° - 14° 19' 26.20164'' ... ^ 30° 40' 33.79836" ... 



CHAPTER II.— PAGE 8. 



CHAPTER II. 

Art. 15. — Page 8. 

3. Here the opposite side = 2, and the adjacent side = 3. There- 
fore the hypotenuse = V2 2 + 3 2 = Vl3. Then, 

sin A — , cot A = -, vers A = 1 , 

Vis 2 _ Vis 

3 a - ^ IS 2 
cos A = — — , sec -* — ~ tt - i covers A — 1 

VI3 1_ Vl3 

2 ' 

2 

4. Sin J.= 1 — covers .A = -• 

5 

Here the opposite side = 2, and the hypotenuse = 5. Therefore the 
adjacent side = V5 2 — 2 2 = V2L Then, 

cos^^^H, cotA = ^, wA = l 

5 2 2 

tan^l = — =z , sec -4 = —^=, vers^l=l— ^-^i. 

V21 V21 5 

5. Here the hypotenuse = 4, and the opposite side = 1. Therefore 
the adjacent side = V4 2 — l 2 = Vl5. Then, 

sin A — -, tan A — , vpr<5 a _ -i Vl5 

4 Vl5 vers^-1 _ f 

cos .4 = ^, cot ^ = VIg ' covers ^1 = §. 

4 .4 4 

sec J. = —y 

V15 
3 

6. Cos A—\ — vers A = — 

4 

Here the adjacent side = 3, and the hypotenuse = 4. Therefore the 
opposite side = V4 2 — 3 2 — y/l. Then, 

sin^=— , cot4 = — , csc^l=— , 

4 V7 Vl 

ta.nA = yl 9 sec.4 = | covers^l=l-^. 

3 3 4 



4 KEY TO ESSENTIALS OF TRIGONOMETRY. 

7. Here the opposite side = x, and the hypotenuse = y. Therefore 
the adjacent side = Vy 2 — x 2 . Then, 



cos 4 = — ^ — , cot A = — ^ — , vers A=l ^ 



tan J. = — * sec ^4 = — ^ covers = 1 — -• 

■\/y 2 — x 2 s/y 2 — x 2 V 

esc A — £ y 
x 

8. Here the hypotenuse = 13, and the adjacent side = 5. Therefore 
the opposite side = Vl3 2 - 5 2 = 12. Then, 

sin A = — , tan A = — , vers 4 = — , 

13 5 13 

cos .4 = — , cot A = — , covers A = — • 

13' 12' 13 

esc A = — , 
12 

9. Here the adjacent side = x, and the opposite side == 1. Therefore 
the hypotenuse = V;r 2 + 1. Then, 

sin A = — , tan A = -, vers A=\- 



y/x 2 + 1 * V* 2 + 1 

cos ^4 = — , sec A = - 1 — , covers J. = 1 • 



V* 2 -t 1 _*_ V* 2 + 1 

esc A = V# 2 + 1, 

10. Here the adjacent side = 8, and the hypotenuse = 17. There- 
fore the opposite side = Vl7 2 — 8 2 = 15. Then, 

sin A=-- y cot A = — , vers A — — , 

17 15 17 

tan A = — , sec A = — , covers A = — • 

8 8 17 

esc A = — , 
15 



11. Here the hypotenuse = Va 2 + b 2 , and the adjacent side = b. 
Therefore the opposite side = Va 2 + b 2 — b 2 = a. Then, 

b 



sin A = — , tan A = -, vers A=l — 



Va 2 + b 2 h Va 2 + & 2 



cos A = — , cot ^ — ~» covers ^L = 1 — - 



Va 2 + 6 2 a Va 2 + b 2 

Va 2 + 6 2 
esc A — ! — , 



CHAPTER III. — PAGE 30. 



CHAPTER III. 

Art. 53. —Page 30. 

3. If A is acute, 450° — A is in the first quadrant. Then, 

sin (450° — A)= cos A, cos (450° — A) = sin A, 

tan (450° - A) = cot A, cot (450° - A) = tan A, 

sec (450° —^4) — esc A, esc (450° — A) = sec A. 

4. If J. is acute, 450° + A is in the second quadrant. Then, 
sin (450° + A) = cos A, cos (450° + A) = — sin A, 
tan (450° + A) = — cot ^4, cot (450° + A) = — tan -4, 
sec (450° + A) = — esc ^4, esc (450° + A) = sec A. 

5. If A is acute, 540° — A is in the second quadrant. Then, 
sin (540° — A) = sin A, cos (540° — A) — — cos -4, 
tan (540° — A) = — tan .4, cot (540° — A) — — cot ^4, 
sec (540° — ^4) = — sec A, esc (540° — A) = esc A. 

6. If -<4 is acute, 540° + A is in the third quadrant. Then, 

sin (540° + A) = — sin A, cos (540° + .4) = — cos A, 

tan (540° + A) = tan ^4, cot (540° + A) = cot .4, 

sec (540° + A) = — sec ^4, esc (540° + A) = — esc J.. 

7. If ^4 is acute, 630° — A is in the third quadrant. Then, 

sin (630° — A) = — cos ^4, cos (630° - .4) = - sin A, 

tan (630° - A) = cot .4, cot (630° — A) = tan .4, 

sec (630° — ^L) = — esc J., esc (630° — A) = — sec A 

8. If -4 is acute, 900° — A is in the second quadrant. Then, 
sin (900° - A) = sin ^4, cos (900° - ^4) = - cos A, 
tan (900° - A) = - tan .4, cot (900° — .4) == — cot ^4, 
sec (900° — A) = - sec ^4, esc (900° - .4) = esc A 

9. If A is acute, — 90° + A is in the fourth quadrant. Then, 
sin (- 90° + A) = - cos 4, cos (— 90° + A) = sin 4, 
tan (— 90° + ^4) = - cot .4, cot (- 90° + .4) = - tan .4, 
sec (- 90° + A) = esc A, esc (- 90° + A) = - sec A 



6 KEY TO ESSENTIALS OF TRIGONOMETRY. 

10. If 4 is acute, — 90° — 4 is in the third quadrant. Then, 
sin (- 90° — 4) = — cos 4, cos (— 90° - A) = — sin A, 
tan (- 90° — A) = cot A, cot (- 90° - A) = tan A, 
sec (— 90° — A) = — esc 4, " esc (- 90° — 4) = — sec A 

11. If 4 is acute, — 180° + A is in the third quadrant. Then, 
sin (- 180° + A) =— sin .4, cos (- 180° + A) = — cos .4, 
tan (- 180° + -4) = tan A, cot (- 180° + A ) = cot 4, 
sec (- 180° + A) = - sec 4, esc (- 180° + 4) = - esc A. 

12. If J. is acute, — 180° — A is in the second quadrant. Then, 
sin (- 180° - A) = sin .4, cos (- 180° - A) = - cos 4, 
tan (— 180° - 4) = — tan 4, cot (- 180° — 4) = - cot 4, 
sec (— 180° — A) == — sec 4, esc (- 180° - 4) = esc 4. 

13. If 4 is acute, — 270° + 4 is in the second quadrant. Then, 
sin (— 270° + 4) = cos 4, cos (— 270° + 4) = — sin 4, 
tan (— 270° + 4) = - cot 4, cot (- 270° -f 4) = — tan 4, 
sec (— 270° + 4) = - esc 4, esc (— 270° + 4) = sec 4. 

14. If 4 is acute, — 720° + 4 is in the first quadrant. Then, 
sin (- 720° + A) = sin 4, cos (- 720° + 4) = cos 4, 
tan (-720° + 4)- tan 4, cot (-720° + 4) = cot 4, 
sec (- 720° + 4) = sec 4, esc (- 720° + 4) = esc 4. 

Art. 54. — Page 30. 

2. cos 152° = cos (180° -28°) =-cos28°; 
or, = cos (90° + 62°) = - sin 62°. 

3. tan 522° = tan (540° — 18°) = — tanl8°; 

or, = tan (450° + 72°) = - cot 72°. 

4. sec (- 77°) = sec (0° - 77°) = sec 77° ; 
or, = sec (- 90° + 13°) = esc 13°. 

5. esc 230° = esc (180° + 50°) =-csc50°; 
or, = esc (270° - 40°) = - sec 40°. 

6. cot (- 129°) = cot (- 180° + 51°) *= cot 51° ; 
or, = cot (- 90° - 39°) = tan 39°. 



CHAPTER III. — PAGES 30, 31. 

7. sin 805° = sin (900° - 35 c ) = sin 35° ; 
or, = sin (810° + 55°) = cos 55°. 

8. cot 83° = cot (90° -7°) = tan 7°. 

9. sin (- 50°) = sin (- 90° + 40°) = - cos 40°. 

10. sec 165° = sec (180° - 15°) = - sec 15°. 

11. cos (- 303°) = cos (- 270° - 33°) = sin 33°. 

12. tan 520° = tan (540° - 20°) = - tan 20°. 

13. esc 768° = esc (810° - 42°) = sec 42°. 

Table. — Page 31. 
Since 120° = 180° - 60°, we have 

sin 120° = sin 60° = -VS. cot 120° = ■ 

2 

cos 120° = - cos 60° = - -• sec 120° = ■ 

2 

tan 120° = - tan 60° = — V§. esc 120° = 
Since 135° = 180° - 45°, we have 



cot 60° = 


-H 


sec 60° = 


-2. 


esc 60° = 


ivs. 



sin 135°= sin 45°= -V2. 
2 


cot 135° = - cot 45° = - 1. 


cos 135° = - cos 45° = - -V2. 
2 


sec 135° = - sec 45° = — V2. 


tan 135° = - tan 45° = -1. 


esc 135° = esc 45° = V2. 


Since 150° == 180° - 30°, we have 




sin 150° = sin 30° = -• 

2 


cot 150° = — cot 30° = — VS. 



cos 150° = - cos 30° = - - V3. sec 150° = - sec 30° = - -VS. 
2 3 

tanl50° = -tan30° = --V3. esc 150°= esc 30°= 2. 
o 



Since 210° = 180° + 30°, we have 

sin 210° = - sin 30° = — - eot 210° = 



cos 210° = - cos 30° = - - V3. sec 210° = 
2 

tan 210°= tan 30°= l\/S. esc 210°: 



cot 30° = 


- Vs. 


sec 30° = 


- 2 V3. 

3 


esc 30° = 


-2. 



; KEY TO ESSENTIALS OF TRIGONOMETRY. 

Since 225° = 180° + 45°, we have 

sin225° = -sin45°=--\/2. cot 225°= cot 45°= 1. 

2 

cos 225° = - cos 45° = - - V2. sec 225° = - sec 45° = - V2. 

2 

tan 225°= tan 45°= 1. esc 225° = - esc 45° = - V2. 



Since 240° = 180° + 60°, we have 

sin 240° = - sin 60° = — - V3. cot 240° = cot 60° = - V3. 
2 3 

cos 240° = - cos 60° = - -- sec 240° = - sec 60° = - 2. 
2 

tan 240° = tan 60° - a/3. csc 240° = - esc 60° = - - V3. 



Since 300° = 360° - 60°, we have 

sin 300° = - sin 60° = - - V3. cot 300° = - cot 60° = - - VS. 
2 3 

cos 300°= cos 60°= 1. sec 300°= sec 60°= 2. 

2 

tan 300° = - tan 60° = - V3. csc 300° = - csc 60° = - - VS. 

3 

Since 315° = 360° - 45°, we have 

sin 315° = - sin 45° = - - V2. cot 315° = - cot 45° = - 1. 

2 

cos 315°= cos45°= i\/2. sec 315°- sec 45°= V2. 

2 

tan 315° = - tan 45° = - 1. csc 315° = - csc 45° = - y/2. 



Since 330° = 360° - 30°, we have 

sin 330° = - sin 30° = - -• cot 330° = - cot 30° ==— VS. 
2 

cos 330°= cos 30°= -V3. sec 330°= sec 30°= ?V3. 
2 3 

tan 330° = - tan 30° = - - y/S. csc 330° = - csc 30° = - 2. 



scissa = ± V4 2 — 


1* = 


r+Vl5. Then, 


a Vl5 
cos A = ± — — , 
4 




cot A = + Vl5, 


tan A— ± , 

Vl5 




A 4 

sec A = ± , 

Vl5 



CHAPTER III. — PAGE 34. 9 

Art. 56. — Page 34. 
3. Here the ordinate = 1, and the distance = 4. Therefore the 



esc A = 4. 



4. Here the abscissa = 2 and the ordinate = 1, or the abscissa = — 2 
and the ordinate = — 1. Therefore the distance = V2 2 + 1 2 = Vd. 
Then, 

sin A = ± , tan A = -, esc A = ± V5. 

V5 2' 

2 /\ 
cos^=±— , sec.A=±— , 

V5 2 

5. Here the distance = 3, and the ordinate = — 2. Therefore the 
abscissa = ± V3 2 - 2 2 = ± V5. Then, 

sin^4 = , tan A=t — •, sec^4 = ± 

3 V5 V5 

A VE . A VE 

cos A = ± , cot ^4 = + — , 

3 2 

6. Here the ordinate = 8 and the abscissa = — 15, or the ordinate 
= — 8 and the abscissa = 15. Therefore the distance = V8 2 + 15 2 = 17. 
Then, 

o -i c -| n 

sin A = ± — , cot A = , esc A = ± — 

17 8 8 



A 15 

cos A=+ — , 




17 

sec A = + — , 
15 


7. Here the 


distance 


= 4, and the al 


ordinate = ± V4 2 


-3 2 =± 


V7. Then, 


sin A = ± — j 
4 




tan A = ± , 

3 


a 3 

cos A = -, 

4 




♦ a 3 
cot A = ± , 

V7 



CSC .4= ± • 

V7 



8. Here the abscissa = — 1, and the distance = 2. Therefore the 
ordinate = ± V2 2 - l 2 = ± V3. Then, 

sin A — ± — » cot A — -=y , esc A — ± 

2 V3 V3 

tanj.= + V3, sec A = —2, 



10 KEY TO ESSENTIALS OF TRIGONOMETRY. 

9. Here the distance == V2, and the ordinate = 1. Therefore the 



abscissa = ± V2 — 1 = 


± 1. Then, 


sin A = , 

V2 


tanJ. = ± 1, sec«4=± V2. 


A 1 

cos A — + — , 

V2 


cot A = ± 1, 


10. Here the ordinate = 2 V2 and the abscissa = 1, or the ordinate 


= — 2 V2 and the abscissa = — 1. Therefore the distance == V8 +1 = 3. 


Then, 




• A 2 ^2 

sin A = ± , 

3 


cot A = , esc A = ± 

2V2 2V2 


A 1 

cos A = ± -, 

3 


sec A = ± 3, 


11. Here the abscissa = — «, and the distance = b. Therefore the 


ordinate = ± V6 2 — a 2 . 


Then, 
cot^4-T a , csc^ = ± . 


V6 2 - a 2 

sin A = ± > 

b 


V& 2 -a 2 V& 2 -a 2 


, . V& 2 - a 2 

tan il = + , 

a 


sec ^1 = — , 
a 


12. Here the ordinate = x, and the distance = 1. Therefore the 


abscissa = ± Vl — x' 2 . 


Then, 




cot A — ± , esc A— — 

X X 


cos A = ± Vl — x 2 , 


tan A — ± , 


A 1 
sec A — ± 



Vl — a: 2 Vl — x 2 

13. Here the abscissa = 1 and the ordinate = x } or the abscissa = — 1 
and the ordinate = —x. Therefore the distance = Vl -f x 2 . Then, 



sinA=:± — x tan^l^ x, csc^L=± i — « 

V1 + * 2 

cos A = ± — . ■ , , sec A = ± Vl -f x 2 , 

Vl + x 2 

14. Here the distance = Va 2 + b 2 , and the abscissa = a. Therefore 
the ordinate = ± Va 2 + b 2 — a 2 = ± b. Then, 

h + A & 

Va 2 + b 2 a 

— , cot A= ±- 

Va 2 + b 2 b 



. A b a b A Va 2 + b 2 

sin^4=± — , t2LnA=±-, cscA = ± ■ — • 

a b 

cos A = a cot A = ± -, 



CHAPTER IV. — PAGE 36. 11 



CHAPTER IV. 

Art. 60. — Page 36. 

sin A = tan A cos A (Art. 59) = ^A = tanA (Art. 53). 

sec J. Vl + tanM 

sin A = — i- - * (Art. 58). 

esc A Vl + eot 2 ^. 



1 vsec 2 .4-1 



sin A = vl — cos-^L = -4/1 — - = ■ — - 

\ sec 2 A sec A 

cosA = —L- = . * (Art. 58). 

sec ^1 Vl + tan 2 A 

cos^ = cot^lsin^ (Art. 59) = 9^lA = cotA (Art. 58). 

esc J. Vl + cot 2 ^ 



cos A = 
tan A = 


Vi- 

sin A 
cos A 

1 


sin' 2 A 


= V 

sin ^4 










tan A = 


Vl 


— sin 2 
1 


A 









1 Vcse 2 4-1 



esc 2 A esc 4. 



cot ^ Vcsc 2 A- I 



= Vl-cos 2 4 
cos A 

(Art. 58). 



Since the cotangent, secant, and cosecant are the reciprocals of the 
tangent, cosine, and sine, respectively, we have : 



cot 4. = 


Vl — sin 2 
sin A 

1 


A 


cos 4 


1 












Vl — cos 2 A 


Vsec 2 A - 
esc A 


-1 


sec A — 


Vl -f cot 2 A _ 
cot J. 






Vl 


— sin 2 
1 


A 


Vcsc 2 A — 
sec A 


■1 


esc A — 


Vl 4- tan 2 A 
tan A 






Vl 


— cos' : 


y A 


Vsec 2 A - 


-1 



12 KEY TO ESSENTIALS OF TRIGONOMETRY. 



CHAPTER V. 

Art. 78. — Pages 51 to 53. 

o sin (a? + 3/) _ sin x cos y + cos x sin y 
sin (a: — y) sin a: cos y — cos a: sin y 

sin a: cos ?/ cos a: sin 3/ 

_ cos x cos 3/ cos x cosy _ tana; -f tan y 

sin x cos y cos a- sin 3/ tan x — tan ?/ 

cos x cos 3/ cos x cos j/ 

^ cos (x + ^) _ cos x cos 3/ — sin x sin # 
cos (a: — ?/) cos x cos 3/ + sin x sin 3/ 

cos a- cos y sin a; sin y 
_ sin a; sin y sin a: sin 3/ cot x cot y -- 1 



cos x cos 3/ sin x sin y cot x cot 3/ + 1 
sin a: sin y sin a: sin y 



e sin (a: + 3/) __ sin a: cos y + cos a: sin y 
cos (a: — y) cos x cos y + sin a; sin y 

sin a: cos y cos a: sin 3/ 
_ sin x cos 3/ sin x cos y _ 1 + cot x tan 3/ 

cos a: cos y .sin a: sin y cot a: -f tan y' 
sin a: cos 3/ cin x cos 3/ 

6. sin (45° + 3/) = sin 45° cos 3/ + cos 45° sin y 

= J_ . cosy + -^ • sin 3/ (Art. 16) =**9+&»9. 

V2 V2 V2 

7. tan(60°-y) = tan 60°- tan y = VS-tany (Art 

l + tan60°tany l+V3tany 

8# sina:+ sin y = 2 sin | (ar + 3/) cos j (x ^ y) = ^ 
cosa:+cosy 2 cos J (a: + y) cos J (a; — y) 2 ^ 

Q sin a? + sin w 2 sin ^ (a? + y) cos J (a: — y) . , N 

9. =? = — ^ . - > ' ^ J x . , v , £Z^ = — cot J (ar — y). 

cos a? — cos 3/ — 2 sin ^ (x + 3/) sin J {x — y) 



CHAPTER V. — PAGE 52. 13 



,,* ^ n* j ro • o 2sinxcosx 

10. By Arts. 74 and 58, sin 2 x = -— — • 

sin" 2 a: + cos 2 x 

Dividing each term of the fraction by cos 2 x, 

2 sin a: 

cos x 2 tan x 



»in2x = 



1 sin 2 a: 1 + tan 2 x 
cos 2 x 



11. esc 2 a: = = = J sec x esc x (Art. 57). 

sin 2 x 2 sin x cos a: 



12. tanx + cotx 



13. cot a: — tana: 



sin a: cos a: _ sin 2 a: + cos 2 x 



cos a: 


sin x 




sin a: cos x 






1 






2 


2 




sin x cos a: 


2 


sin a: cos x 


sin 2 


X 


cotx — 


1 
cota: 


= 


cot 2 a: — 1 
cota: 







= 2 ( cot2x - 1 \ := 2 cot 2a: (Art. 74). 
V 2 cot a: J V y 



14 (1 + tanx) 2 - (1-tanx) 2 
(1 + tan x) 2 + (1 — tan a:) 2 



1 + 2 tan a: + tan 2 a: — 1 -f- 2 tan x — tan 2 a: 

1 + 2 tan x + tan 2 x + 1 — 2 tan a- + tan 2 a: 

4 tan a: 2 tan x 



2 + 2tan 2 x l + tan 2 x 



= sin 2 x, by Ex. 10 



15. sin (a: + 3/) sin (x — 3/) 

= (sin x cos 3/ + cos a- sin y) (sin x cos 3/ — cos x sin 3/) 

= sin 2 x cos 2 y — cos 2 x sin 2 y 

— sin 2 x (1 — sin 2 y) — (1 — sin 2 x) sin 2 y 

= sin 2 x — sin 2 3/. 

16. cos (x + 3/) cos (x — y) 

= (cos x cos 3/ — sin x sin 3/) (cos x cos 3/ + sin x sin 3/) 

= cos 2 x cos 2 3/ — sin 2 x sm 2 y 

= cos 2 x cos 2 y — (1 — cos 2 x) (1 — cos 2 3/) 

= cos 2 x cos 2 y — 1 + cos 2 x + cos 2 y — cos 2 x cos 2 3/ 

== cos 2 x — (1 — cog 2 3/) == cos 2 x — sin 2 3/. 



14 KEY TO ESSENTIALS OF TRIGONOMETRY. 

-,« 9 9 1 sin 2 x + cos' 2 x 

17. sec 2 x esc 2 x = — — - = . — 

cos 2 x sin 2 x cos 2 x sin 2 x 

sin 2 a: . cos 2 a- 1 | 1 

+ — n r^r= — r- +■ 



cos 2 x sin 2 a: cos 2 a: sin 2 x cos 2 # sin 2 x 

= sec 2 a: + esc 2 x. 

• 

18. cos y + cos (120° + y) + cos (120° - y) 

— cos 3/ + cos 120° cos y — sin 120° siny + cos 120° cos y + sin 120° siny 
= cos y + 2 cos 120° cos y = cos 3/ — cos 3/ (Art. 55) = 0. 

19. sin A sin (JB - C) + sin ^ sin(C- A) + sin Csin (J. - B) 

= sin ^1 (sin I? cos C — cos 5 sin C) + sin Z?(sinCcos-4 — cosOsin^l) 
+ sin C (sin A cos jB — cos A sin _B) 

= sin A sin 5 cos C — sin J. cos Z? sin C + sin 5 sin C cos J. 

— sin B cos O sin J. + sin C sin J. cos B — sin (7 cos A sin 2? = 0. 

20. cos (4 + B) cos (4 - .B) + cos (J5 + C) cos (5 - C) 

+ cos(C+^l)cos(C-^) 
= cos 2 J. — sin 2 B + cos 2 B — sin 2 (7 + cos 2 O — sin 2 4, by Ex. 16, 
= cos 2^1 + cos2J9+cos2C (Art. 74). 

<yi cos x — cos 3 x _ — (cos 3 a: — cos x) 
sin 3 a: — sin x sin 3 x — sin x 

2 sin A (3 x + a:) sin £ (3 x — x) 

— — — — - = — - = tan 2 x, 

2 cos J (3 x + a?) sin J (3 x — x) 

22 cos 80° + cos 20° _ 2 cos j (80° + 20°) cos j (80° - 20°) 
sin 80° - sin 20° ~ 2 cos J (80° + 20°) sin J (80° - 20°) 
= cot 30° = V3 (Art. 17). 

23. sin (x + y + 2) = sin [(a? + ?/) + 2] 
= sin (a? + 3/) cos 2 + cos (a: + y) sin 2 

= (sin x cos ?/ + cos x sin ?/) cos 2 + (cos a: cos y — sin a? sin ?/) sin 2 
= sin a? cos ?/ cos 2 + cos x sin 2/ cos z + cos a- cos y sin 2 — sin x sin j/ sin z. 

24. cos (a: + y + 2) = cos [(a- + y) + 2] 
= cos (a? + 3/) cos z — sin (a: + y) sin 2 

=? (cos x cos 3/ — sin x sin 3/) cos 2 — (sin a: cos y + cos a: sin y) sin 2 
= cos x cos 3/ cos 2 — sin x sin 3/ cos 2 — sin ar cos y sin 2 — cos a? sin y sin 2. 



CHAPTER V. — PAGE 53. 15 

25. sin 3 x = sin (2 x + x) — sin 2 x cos x + cos 2 a? sin x 

= 2 sin x cos 2 x + (1—2 sin 2 a?) sin a; (Art. 74) 
= 2 sin x (1 — sin 2 a?) + sin x — 2 sin 3 a? 
= 3 sin x — 4 sin 8 a\ 

26. cos 3 x = cos (2i + x) = cos 2 a: cos a: — sin 2 x sin x 

= (2 cos 2 x — 1) cos x — 2 sin 2 x cos a? (Art. 74) 
= 2 cos 3 x — cos x — 2 (1 — cos 2 a;) cos a: 
= 4 cos 3 a; — 3 cos, a\ 

27. tan3:r = tan(2:r + :r)= tan2 *+*anx 



1 — tan 2 a: tana? 

2 tana: 



• + tan x 



}-***** (Art. 74) 

.. __ 2 tan 2 x 



1 — tan 2 a: 
2 tan x -f tan x — tan 3 a: _ 3 tan x — tan 3 a? 



1 — tan 2 x — 2 tan 2 a: 1—3 tan 2 a? 

28. sin (2 x + y) — 2 sin .r cos (a: + 3/) 
= sin [ (x + ?/) + x~\ — 2 sin x cos (a: + ?/) 
= sin (a: + ?/) cos x + cos (a? + ?/) sin a? — 2 sin a? cos (a: + y) 
= sin (a: + y) cos a; — cos (a: + #) sin x 
= sin [ (a; + y) — x~] = sin y. 

oq sin 3 a? __ cos 3 a? _ sin 3 x cos x — cos 3 x sin a: 
sin a; cos x sin a: cos x 

= »fa(8*-*) (Ar t.74) = 2. 
J sin 2 x 

30. 1 + cos 2 a cos 2 3/ = 1 + (2 cos 2 x — 1) (1 — 2 sin 2 y) 
= 2 cos 2 a: + 2 sin 2 y — 4 cos 2 a: sin 2 3/ 

= 2 cos 2 x (sin 2 ?/ + cos 2 y) + 2 sin 2 3/ (sin 2 a: + cos 2 a:) — 4 cos 2 x sin 2 3/ 
= 2 (sin 2 a: sin 2 ?/ + cos 2 x cos 2 3/) . 

01 -1 . x x o -1 1 sin x sin 2 a: 

31. 1 + tana: tan 2 a: = 1 -\ 

cos a: cos 2 a: 

cos 2 x cos x 4- sin 2 a: sin a: cos (2 a: — a:) 1 «-> 

= ! — - — i i = =r sec 2 X. 

cos 2 x cos a; cos 2 ar cos a: cos 2 a: 



16 KEY TO ESSENTIALS OF TRIGONOMETRY. 

32. sin 4 x = 2 sin 2 a: cos 2 a: = 4 sin x cos a:(l — 2 sin 2 a;) (Art. 74) 

= 4 sin a: cos x — S sin 3 a: cos a\ 

33. cos4a; = 2cos 2 2a:- 1 = 2(2cos 2 a?- I) 2 - 1 (Art. 74) 

= 8 cos 4 x — 8 cos 2 x + 2 — 1 = 8 cos 4 a: — 8 cos 2 a: + 1. 

34. sin 5 x = sin (4 x + a?) = sin 4 a? cos a: + cos 4 a: sin a* 

= 4 sin a: cos 2 x — 8 sin 3 a: cos 2 a; -f 8 cos 4 a: sin x — 8 cos 2 a: sin a: 

+ sina; (Exs. 32, 33) 
= 4 sin a: (1 — sin 2 x) — 8 sin 3 a: (1 — sin 2 ar) + 8 (1 — sin 2 x) 2 sin a: 

— 8 (1 — sin 2 x) sin a: + sin x 
= 4 sin x — 4 sin 3 a: — 8 sin 3 a: + 8 sin 5 a; + 8 sin x — 16 sin 3 a: 

-f 8 sin 5 x — 8 sin x + 8 sin 3 x + sin a: 
= 5 sin x — 20 sin 3 x + 16 sin 5 a:. 

35. sin 15° = sin (45° - 30°) = sin 45° cos 30° - cos 45° sin 30° 

= iV§- JV3-1V2 • J = i(V6-V2) = cos75°. 
cos 15° = cos (45° - 30°) = cos 45° cos 30° + sin 45° sin 30° 
= J V2 • JV3 + iV2 • i = I ( V6 + V2) = sin 75°. 

36. tan 15° = 1 "" cos 80 ° = esc 30° - cot 30° = 2 - V3 = cot 75°. 

sin 30° 

cot 15° = 1 + cos 80 ° = esc 30° + cot 30° = 2 + V3 = tan 75°. 
sin 30° 

37. sin 220 30' = JIE^iE^ S = ,B 

\ 2 V 2 ^4 

= jVi - V2. 

cos 220 30' = ^1 + co. 45Q = ^jg^ ^Gt_^ 



= i V2 + V2. 



38. tan 22° 30' = 1 - cos 45 ° _ csc 450 _ cot 45° = V2 - 1. 

sin 45° 

cot 22° 30' = 1 + CQS ^ 5 ° = csc 45° + cot 45° = V2 + 1. 

sin 45° 



CHAPTER VI. — PAGE 57. 17 



CHAPTER VI. 

Art. 91. — Page 57. 

2. log 6 = log (2x3) = log 2 + log 3 = .3010 + .4771 = .7781. 

3. log 14 = log (2x7) = log 2 + log 7 = .3010 + .8451 = 1.1461. 

4. log 8 = log(2 x2x2) = log2 + log2 + log2 = 31og2 

= 3 X .3010 = .9030. 

5. log 12 = log(2x2x3) = log2 + log2 + log3 

= 2 log 2 + log 3 = .6020 + .4771 = 1.0791. 

6. log 15 = log (3x5) = log 3 + log 5 =.4771 + .6990 =1.1761. 

7. log 21 = log (3x7)= log 3 + log 7 = .4771 + .8451 = 1.3222. 

8. log 63 = log(3x3x7) = log3 + log3 + log7 = 21og3 + log7 

= .9542 + .8451 = 1.7993. 

9. log 56 = log(2x2x2x7) = 31og2 + log7 

= .9030 +.8451 = 1.7481. 

10. log 84 = log(2x2X3x7) = 21og2 + log3 + log7 

= .6020 + .4771 + .8451 = 1.9242. 

11. log 45 = log (3x3x5) = 2 log 3 + log 5 =.9542+. 6990 =1.6532. 

12. log 98 = log (2x7 X7) = log 2 + 2 log 7 = .3010 +1.6902 =1.9912. 

13. log 105 = log (3 X 5 X 7) = log 3 + log 5 + log 7 

= .4771 + .6990 + .8451 = 2.0212. 

14. log 112 = log (2 X 2 X 2 X 2 x 7) = 41og2 + log7 

= 1.2040 + .8451 = 2.0491. 

15. log 144 = log (2 X 2 X 2 X 2 X 3 X 3) = 41og2 + 21og3 

= 1.2040 + .9542 = 2.1582. 

16. log216 = log (2 x 2 X 2 X 3 X 3 x 3) = 31og2 + 31og3 

= .9030+1.4313 = 2.3343. 



18 KEY TO ESSENTIALS OF TRIGONOMETRY. 

17. log 135 =r log (3 x 3 X 3 X 5) = 3 log 3 + log5 

= 1.4313 + .6990 = 2.1303. 

18. log 168 = log (2 x 2 x 2 X 3 x 7) = 31og2 + log3 + log 7 

= .9030 + .4771 + .8451 = 2.2252. 

19. log 147 = log (3 x 7 X 7) = log 3 + 2 log 7 = .4771 + 1.6902 = 2.1673. 

20. log 375 = log (3 X 5 X 5 X 5) = log 3 + 31og5 

= .4771 + 2.0970 = 2.5741. 

21. log 343 = log (7 X 7 X 7) = 3 log 7 = 2.5353. 

Art. 93. — Page 58. 

2. log - = log 7 - log 3 = .8451 - .4771 = .3680. 

o 

3. log — = log 10- log7 = l-. 8451 = .1549. 

4. log 3J = log — = log 10 -log 3=1 -.4771 = .5229. 

o 

5. log 35 = log — = log (10x7) -log 2 = log 10 + log 7 -log 2 

2 

= 1 + .8451 - .3010 = 1.5441. 

91 

6. log ^=log21-logl6 = log(3x7)-log(2x2x2x2) 

lb 

= log 3 + log 7 - 4 log 2 = .4771 + .8451 - 1.2040 = .1182. 

7. logl25 = log(5x5x5) = 31og5. 

log 5 = log — = 1 - log 2 = 1 - .3010 = .6990. 
.-. log 125 = 3 X .6990 = 2.0970. 

8. log ^ = log(2x3x7)-log(5x5) = log24-lo g 3 + log7-21og5. 

25 

log 5 = log — = 1 - log 2 = .6990. 

A 

... log — = .3010 + .4771 + .8451 - 1.3980 = .2252. 
8 25 



9. log 175 = log (5 X 5 X 7) = 2 log 5 + log 7. 

log 5 = log — = 1 - log 2 = .6990. 
\ log 175 = 1.3980 + .8451 = 2.2431. 



CHAPTER VI. — PAGES 58, 59. 19 

10. log 11 § = log !—■ = log 100 - log (3 X 3) 

= 2-2 log 3 = 2- .9542 = 1.0458. 

11. log 7f = log — = log i°5 = log 100 -log (2x7) 

= 2 - log 2 - log 7 = 2 - .3010 - .8451 = .8539. 

12. log ?5 = log(5x7)-log(2x3)=log5 + log7-log2-log3. 

6 

log 5 = log — = 1 - log 2 == .6990. 

... log ?& = .6990 + .8451 - .3010 - .4771 = .7660. 
6 

13. log 5f = log^ = log(7x7)-log(3x3)=21og7-21og3 

= 1.6902 -.9542 =.7360. 

Art. 96. — Page 59. 

3. log 3* = ?log3=-X.4771 = .2863. 

5 5 

4. log 2 9 = 9 log 2 = 9 X. 3010 = 2.7090. 

5. log 7 5 = 5 log 7 = 5 X. 8451 = 4.2255. 

i i 

6. log 5 5 = -log5. 

5 

log 5 = log ™ = 1 -log 2 = . 6990. 

.,lo g 5 * = «=.1398. 
* 5 

7. log 123 = - ^g 12. 

log 12 = log (2 x 2 x 3) = 2 log 2 + log 3 = .6020 + .4771 = 1.0791. 

/. log 12* = ? x 1.0791 = .7194. 
o 

8. log 21^ = - log 21. 

A 

log 21 = log (3 X 7) = log 3 + log 7 = .4771 + .8451 = 1.3222. 
.-. log 21^=1^? =.6611. 



20 KEY TO ESSENTIALS OF TRIGONOMETRY. 

9. log 14* = 4 log 14. 

log 14 = log (2x7) = log 2 + log 7 = .3010 + .8451 = 1.1461. 
.-. log 14 4 = 4 X 1.1461 = 4.5844. 

10. log 253 = 1 log 25. 

o 

log 25 = log — = logl00-log(2x2) 
= 2-2 log 2 = 2- .6020 = 1.3980. 

/. log 253 = 1 x 1.3980 = 3.2620. 
o 

11. log 15* = -log 15. 

6 

log 15 = log (3 X 5) = log 3 + log 5. 

log 5 = log — =1 -log 2 =.6990. 

.-. log 15 = .4771 + .6990 = 1.1761. 

.-. log 15* = - X 1.1761 = .9801. 
6 

12. log V7 = !2£l = :M51 = .4225. 

B 2 2 

13. log ^3 = 1^= ill! =.1590. 

5 3 3 

14. log ^ = l2S2 = ^010. = 0430 

* 7 7 

15. log ^ = !2£* 

5 6 

log 5 = log — = 1 - log 2 = .6990. 

, 6/v .6990 11A - 

.-. log v o = = .1165. 

5 6 

16. log</35 = ^. 

4 

log 35 = log (5x7) = log 5 + log 7. 

log 5 = log — = 1 - log 2 = .6990. 
:. log 85 = .6990 + .8451 = 1.5441. 
... log </35 = !^i=. 3860. 



CHAPTER VI. — PAGE 59. 



21 



17. log ^98 = ^|5§. 

log 98 = log (2 X 7 2 ) = log 2 + 2 log 7 = .3010 + 1.6902 = 1.9912. 



log ^98 = 



' = .2212. 



18. log^l26 = ^£l26. 

log 126 = log (2 x 3 2 x 7) = log2 + 21og3 + log7 
= .3010 + .9542 + .8451 = 2.1003. 

,.log ^126 = ^P -.1750. 



20. log 



m 



12 

5 = 5 log — = 5 (log 10 - log 3) = 5 (1- .4771) 
o 

= 5 X .5229 = 2.6145. 



21. log ^ = log7*-log5* = ;jlog7-?log5 

5 # 4 6 

= .6338 - .4660 = .1678. 

22. log (3* X 2*) = log 3* + log 2* = i log 3 + § log 2 

o 5 

= .0795 + .1806 = .2601. 

23. log 3^ / 7 = log3 + logv / 7 = log3f 1°|Z 

= .4771 + .2113 = .6884. 

24. log ^log^**^ 

25.log p- = log¥i-logM = 1 ^- 1 ^ 

■s/2 o 5 

= .2817 - .0602 = .2215. 



= .1840. 



26. log ^f = ^° g f= l0g28 ,- l0g5 
log 



5 3 ° 5 3 

28 = log (22 x 7) = 2 log 2 + log 7 = .6020 + .8451 = 1.4471. 



, 3 28 1.4471 -.6990 OAQA 

•'• log Ys" 3 = * 2494 ' 

27. log ^f = logV42-logl0* = ^^-?logl0. 
10 3 2 3 

log 42 = log (2 X 3 X 7) = log2 + log3 + log7 
_ = .3010 + .4771 + .8451 = 1.6232. 

... log 2^ = .8116 - .6667 = .1449. 
10* 



2. 


log 
log 


3. 


log 




log 


4. 


log 




log 


5. 


log- 
log 


6. 


log 
log 



22 KEY TO ESSENTIALS OF TRIGONOMETRY. 



Art. 98. — Page 60. 

18 = log (2 X 3 2 ) = log 2 + 2 log 3 = .3010 + .9542 = 1.2552. 
1.8 = 0.2552. 

225 = log (3 2 X 5 2 ) = 2 log 3 + 2 log 5 

= .9542 + 1.3980 = 2.3522. 
2.25 = 0.3522. 

196 = log (2 2 X 7 2 ) = 2 log 2 + 2 log 7 

= .6020+1.6902 = 2.2922. 
.196 = 9.2922 - 10. 

48 = log (2* x 3)= 4 log 2 + log 3 = 1.2040+ .4771 = 1.6811. 
.048 = 8.6811 - 10. 

384 = log (2 7 x 3) = 7 log 2 + log 3 = 2.1070 + .4771 = 2.5841. 
38.4=1.5841. 

7. log 54 = log (2 x 3 3 ) = log 2 + 3 log 3 = .3010 + 1.4313 = 1.7323. 
.-. log .0054 = 7.7323 - 10. 

8. log 315 = log(3 2 x5x7) = 21og3 + log5 + log7 

= .9542 + .6990 + .8451 = 2.4983. 
.♦. log .000315 = 6.4983 - 10. 

9. log 735 = log(3x5x7 2 ) = log3+log5 + 21og7 

= .4771 + .6990 + 1.6902 = 2.8663. 
.-. log 7350 = 3.8663. 

10. log 405 = log (3* x 5) = 4 log 3 + log 5 

= 1.9084 +.6990 = 2.6074. 
.-. log 4.05 = 0.6074. 

11. log 448 = log (26x7) = 6 log 2 + log 7 

= 1.8060 +.8451 = 2.6511. 
.-. log .448 = 9.6511 - 10. 

12. log 3024 = log (2* X 3 3 X 7) = 4 log2 + 3 log3 + log7 

= 1.2040 + 1.4313 + .8451 = 3.4804. 
.-. log 302.4 = 2.4804. 

13. log 6174 = log(2x3 2 x7 3 ) = log2 + 21og3 + 31og7 

= .3010 + .9542 + 2.5353 = 3.7905. 
.-. log .06174 = 8.7905 - 10. 



CHAPTER VI. — PAGES GO, 65. 23 



14. log (8.1) 7 =71og8.1. 

log 81 = log 3* = 4 log 3 = 1.9084. 
.-. log 8.1 = 0.9084. 

.-. log (8.1) 7 = 7 x .9084 = 6.3588. 

i c i b /cTa log 9.6 

15. log v9.6 = — s 

5 

log 96 = log (2 5 X 3) = 5 log 2 + log 3 
= 1.5050 + .4771 = 1.9821. 
/. log 9.6 = 0.9821. 
.,l 0g ^6 = 5^1 = 1964> 



16. log(22.4)* = - log 22.4. 
8 

log 224 = log (25 x 7) = 5 log 2 + log 7 = 1.5050 + .8451 = 2.3501. 

/. log 22.4 = 1.3501. 



log(22.4)* = M521 _ §1688i 
8 



Art. 105. — Pages 65 and 66. 

1. log (9.238 X .9152) = log 9.238 + log .9152. 
log 9.238 = 0.9656 

log .9152 = 9.9615-10 

0.9271 = log 8.454. 

2. log (130.36 X .08237) = log 130.36 + log .08237. 
log 130.36 = 2.1151 

log .08237 = 8.9157 - 10 

1.0308 = log 10.73. 

3. log (721.3 X 3.0528) = log 721.3 + log 3.0528. 
log 721.3 = 2.8581 

log 3.0528 = 0.4847 

3.3428 = log 2202. 
Result, - 2202. 

4. log (4.3264 x .050377) = log 4.3264 + log .050377. 
log 4.3264 = 0.6361 

log.050377 = 8.7022 — 10 

9.3383 - 10 = log .2179. 



24 KEY TO ESSENTIALS OF TRIGONOMETRY, 

5. log (.27031 X .042809) = log .27031 + log .042809. 
log .27031 = 9.4319-10 

log .042809 = 8.6315-10 

8.0634 - 10 - log .01157. 

6. log (.063165 X 11.134) = log .063165 + log 11.134. 
log .063165 = 8.8005-10 

log 11.134=1.0466 

9.8471 - 10 = log .7032. 
Result, - .7032. 

7. log ^L?- = log 401.8 -log 52.37. 

log 401.8 = 2.6040 
log 52.37 = 1.7191 

0.8849= log 7.672. 

8. log 1^1 = log 7.2321 -log 10.813. 

6 10.813 e 6 

log 7.2321 = 0.8592 
log 10.813 = 1.0339 



9.8253 - 10 = log .6688. 

9. log -^i = log .3384 - log .08659. 
B .08659 B B 

log .3384 = 9.5294-10 

log .08659 = 8.9374 - 10 

0.5920 = log 3.908. 

Result, - 3.908. 

10 - log -r|^r = log 9.163 -log .0051422. 
.0051422 

log 9.163 = 0.9620 

log .0051422 = 7.7112 -10 

3.2508 = log 1782. 

11. log ??^1§ = log 22518 -log 64327. 



log 22518 = 4.3525 
log 64327 = 4.8084 



9.5441 - 10 = log .3500. 



CHAPTER VI. -PAGE 65. 25 



12. log ^m* _ log .007514 - log .015822. 
.015822 



log .007514=7.8758-10 
log .015822 = 8.1993-10 

9.6765 - 10 = log .4748. 
Result, - .4748. 
3.3681 



13. log 

6 12.853 X .6349 

= log 3.3681 + colog 12.853 + colog .6349. 

log 3.3681 = 0.5274 
colog 12.853 = 8.8910 - 10 
colog .6349 = 0.1973 

9.6157 - 10 = log .4127. 

14 i oc 15.008 X .0843 

* .06376 X 4.248 
= log 15.008 + log .0843 + colog .06376 + colog 4.248. 

log 15.008 = 1.1763 

log .0843 = 8.9258-10 
colog .06376 = 1.1955 
colog 4.248 = 9.3718-10 

0.6694 = log 4.671. 

Result, -4.671. 

15 l0 2563 X. 03442 

5 714.8 X. 511 
= log 2563 + log .03442 + colog 714.8 + colog .511. 

log 2563 = 3.4087 

log .03442 = 8.5368 - 10 
colog 714.8=7.1458-10 
colog .511 = 0.2916 

9.3829 - 10 = log .2415. 

1C , 121.6x9.025 . ; o -- 

16. log — = loff 121.6 

S 48,3 X 3662 X =0856 5 

+ log 9.025 + colog 48.3 + colog 3662 + colog .0856. 

log 121.6 = 2.0850 

log 9.025 = 0.9554 
colog 48.3 = 8.3161 - 10 
colog 3662 = 6.4363 - 10 

colog .0856 = 1.0675 

8.8603 - 10 = log .0725. 
Result, - .0725. 



26 KEY TO ESSENTIALS OF TRIGONOMETRY. 

17. log (23.86)3 = 3 X log 23.86. 
log 23.86 = 1.3777 
3 



4.1331 = log 13587. 



18. log (.532) 8 = 8 X log. 532. 
log .532 =9.7259-10 
8 



7.8072 - 10 = log .006415. 



19. log (1.0246) 7 = 7 X log 1.0246. 
log 1.0246 =0.0105 

7 



0.0735 = log 1.184. 
Result, - 1.184. 

20. log (.09323) 5 = 5 X log .09323. 
log .09323 =8.9695-10 
5 



21. 


log5 3 = 


4.8475 - 

: - lOg 5. 

3 e 


- 10 = lo^ 




log 5 = 


: 0.6990; X- = 


0.4660 


22. 


log (.8) 


2 2 

5 = ^log.8. 



log 2.924. 




log .8 


= 9.9031 - 10 

2 





5 )49.8062 - 50 

9.9612 - 10 = log .9146. 

23. log (3.16)^ = -log 3.16. 
o 

log 3.16 =0.4997; X^= 0.6663 
o 



= lo<? 4.638. 



24. log (.021)*=- log .021. 

log .021 =8.3222-10 
5 



2)11.6110-20 

5.8055 - 10 = log .0000639. 



CHAPTER VI. — PAGES 05, 66. 27 

25. logV2 = -log2. 

A 

log 2 = 0.3010; -2 = 0.1505 

= log 1.414. 

26. log y/l = - log 5. 

4 

log 5 = 0.6990; -4 = 0.1747 

= log 1.495. 

27. log #3 = hog 3. 

5 

log 3 = 0.4771; -f- 5 = 0.0954 

= log 1.246. 
Result, — 1.246. 

28. log V^4294 = - log .4294. 

log .4294 = 19.6329 - 20 ; -f- 2 = 9.8164 - 10 

. = log .6553. 



29. log #02305 = ± log .02305. 

o 

log .02305 = 28.3626 - 30 ; -^ 3 = 9.4542 - 10 

= log .2846. 

30. log ^1000 = - log 1000. 

8 

log 1000 = 3 ; -f- 8 = 0.3750 = log 2.372. 

31. log #00951 = - log .00951. 

log .00951 = 67.9782 - 70 ; - 7 = 9.7112 - 10 

= log .5142. 
Result, - .5142. 



1 



32. log #0001011 = - log .0001011. 
5 

log .0001011 = 46.0047 - 50 ; - 5 = 9.2009 - 10 

= log .1588. 

35. log (2* X S^ = - log 2 + | log 3. 

log 2 = .3010; X--.4515 

A 

log 3 =.4771; X-=.3181 

o 

.7696 = log 5.883. 



28 KEY TO ESSENTIALS OF TRIGONOMETRY. 



36. log^ = flog3-?lo g 4. 

43 y 6 

log 3 = .4771 ; X - = .2982 

8 

log 4 =.6021; X-= .4014 

o 



- 10 = log .7885. 



37. log — = -log5-?logl0. 

1Q9 ' ^ 

log 5 =.6990; X - = .2996 





log 


10 = 


l- x 2 - 

' X 9 


= .2222 




38. 


log/ 


6\f _5 

7J 2 


(log 6- 


.0774 = 
log 7). 


= log 1.195. 




log 6 
log 7 


= .7782 
= .8451 

9.9331 


-10 
5 







2 )19.6655-20 

9.8327 - 10 = log .6803. 

39. log ^|)*= ? (log 35 -log 113). 

log 35=1.5441 
log 113 = 2.0531 

9.4910 - 10 
3 



8 )78.4730-80 

9.8091 - 10 = log .6443. 

40. log (^f^y = § Gog .08726 - log .1321). 

log .08726 = 8.9408 - 10 
log .1321 = 9.1209 - 10 

9.8199 - 10 
5 



3 )29.0995-30 

9.6998 -10 = log .5010. 



CHAPTER VI. — PAGE 66. 



29 



41. log^|| = |(log21-logl3). 

log 21= 1.3222 
log 13 =1.1139 

8 ) .2083 

.0260 = log 1.062. 



42. logvf=l(log3-log7). 

> i 9 



log 3, 

lOg 7 : 



.4771 

.8451 



43. 



9)89.6320-90 




9.9591 - 10 = log .9102. 




Result, - .9102. 




l0g (A3-^) 




= | (log 2 - log 3) - 1 (log 3 - log 5). 

o 




log 2= .3010 log 3= .4771 




log 3= .4771 log 5= .6990 




5)49.8239-50 3)29.7781- 


-30 


9.9648 - 10 9.9260 - 


-10 


9.9260 - 10 





.0388 



= log 1.093. 



44. log(V2x^3x VIJ1) 

= hog2+hog3 + llog.01. 
8 o / 

log 2 = .3010; -8= .0376 

log 3 =.4771; --5= .0954 

log .01 = 68 - 70 ; -f- 7 = 9.7143 — 10 



9.8473 -10 = log .7035. 



45. log A 



3258 



= - (log 3258- log 49309). 
49309 5 V S * J 



log 3258 = 3.5129 
log 49309 = 4.6929 



5)48.8200-50 

9.7640 - 10 = log .5807. 



30 KEY TO ESSENTIALS OF TRIGONOMETRY. 



46. log(^V=A(io g 31.63-log429). 



log31.63 = 1.5001 
log 429 = 2.6325 

8.8676 - 10 
3 



17 )166.6028-170 

9.8002-10 = log .6313. 
Result, - .6313. 

47. log 10 ° 3 3 = ? log 100 - - log .7325. 

(.7325)7 3 7 

log 100 = 2; X- = 1-8338 

o 

log .7325 = 9.8648 - 10 

3 . 

69.5944 - 70 -T- 7 = 9.9421 - 10 

1.3912 = log 24.62. 

48. log x/ - 0001289 = - log .0001289 - 1 log .0008276. 

V.0008276 3 4 

log .0001289 = 26.1103 - 30 ; -*- 3 = 8.7034 - 10 
log .0008276 = 36.9178 - 40 ; -r- 4 = 9.2294 - 10 

9.4740 - 10 
= log .2979. 

5 

49. log C 7469 )' = 5 log .7469 - - log .2345. 

(.2345)5 3 2 

log .7469 = 9.8732 - 10 log .2345 = 9.3701 - 10 

5 7 

3 )29.3660 - 30 2 )15.5907-20 

9.7887-10 7.7953-10 
7.7953 - 10 



1.9934 = log 98.50. 

50. log ^-Q 073 = JL log .0073 - - log .68291. 

& 5 1 1 & O ° 



(.68291)2 " 
log .0073 =107.8633 -110 
ividing by 11, = 9.8058 - 10 
9.5857 - 10 


2 

log .68291 = 9.8343 - 10 

5 

2)19.1715-20 


.2201 
= log 1.660. 


9.5857 - 10 



CHAPTER VI. — PAGE 66. 31 



K1 . V5.955X V(;i.2 
51. log 

V298.54 

= I log 5.955 + - log 61.2 + - colog 298.54. 
2i o o 

2 = 0.3874 

3 = 0.5956 
5 = 9.5050 - 10 



log 5.955= 0.7748 ; 

log 61.2= 1.7868 
colog 298.54 = 47.5250 - 50 ; 



0.4880 = log 3.076. 



52. log (538.2 X .0005969)* = - (log 538.2 + log .0005969). 

8 

log 538.2= 2.7310 
log .0005969= 6.7759-10 

8 )79.5069-80 

9.9384 - 10 = log .8678. 

53. log [(18.9503)11 X (.l) 14 ] 

= 11 X log 18.9503 + 14 X log .1. 
log 18.9503 = 1.2777 ; X 11 = 14.0547 
log .1 = 9-10; X 14= 6. -20 

.0547 = log 1.134. 

54. log ^3734.9 X .00001108 =- (log 3734.9 + log .00001108). 

log 3734.9= 3.5723 
log .00001108 = 5.0445 - 10 

6)58.6168-60 

9.7695 - 10 = log .5881. 



55. log [(2.6317)' X (.71272)*] 

= -log 2.6317 + -log .71272. 
4 5 



log 2.6317= .4203 
3 


log .71272= 9.8529-10 

2 


4)1.2609 
.3152 
9.9412 - 


5)49.7058 - 50 
9.9412 - 10 
-10 


.2564 


= log 1.805. 



32 KEY TO ESSENTIALS OF TRIGONOMETRY. 



^.008193 X (-06285)* 
' . .98342 

= -log .008193 + -log .06285 + colog .98342. 
o A 

log .008193 = 27.9134 - 30 ; -r- 3 = 9.3045 - 10 
log .06285= 8.7983-10 
3 



16.3949 - 20 ; h- %= 8.1974 - 10 
colog .98342 = 0.0072 

7.5091 - 10 
= log .003229. 



57. log ( VXJ35 X V.62G67 X V^.0072103) 

= - log .035 + - log .62667 + - log .0072103. 
2 6 3 



log .035 = 18.5441 - 20 
log .62667 = 59.7971-60 
log .0072103 = 27.8579 - 30 



2 = 9.2720 - 10 
-f- 6 = 9.9662 - 10 

3 = 9.2860 - 10 



8.5242 - 10 
= log .03344. 



CHAPTER VII. — PAGES 69, 70. 33 



3. 



CHAPTER VII. 

Art. 110. — Pages 69 to 72. 

a = csmA. b — c cos A. 

log c = 1 .0492 log c = 1 .0492 

log sin A = 9.8378 log cos A = 9.8606 

log a = 0.8870 log 6 = 0.9098 

/. a = 7.708. .-. 6 = 8.124. 

b = a tail 1?. 



cos 5 

log a = 2.8629 log a = 2.8629 

log tan B = 0.4121 log cos B = 9.5576 

log b = 3.2750 log c = 3.3053 

.-. 6=1883. .-. c = 2019.5, 



tan B sin B 

log 6 = 1.6785 log 6 = 1.6785 

log tan B = 0.2916 log sin B = 9.9496 

log a =1.3869 log c = 1.7289 

.-. a = 24.37. /. c = 53.56. 



sin A = -• 6 = 



c tan .A 

log a = 9.7952 log a = 9.7952 

log c = 9.9590 log tan A = 9.9742 

log sin A = 9.8362 log 6 = 9.8210 

.-. A = 43° 17.9'. .-. 6 = .6622. 



5 a u a 

tan A = — c = — 

6 sin ^4 

log a = 0.6990 log a = 0.6990 
log 6 = 0.3010 log sin A = 9.9678 

log tan A = 0.3980 log c = 0.7312 

/. A = 68° 12.2'. .-. c= 5.385. 



34 KEY TO ESSENTIALS OF TRIGONOMETRY. 

6. 6 = - 



b = 


a 
tan A 


log a = 


: 2.1995 


log tan A = 


: 9.9752 



tan ^4 sin. A 

log a =1.9212 log a =1.9212 

log tan A = 0.4912 log sin A = 9.9785 

log b =1.4300 log c = 1.9427 

.-. 6 = 26.91. .-. c = 87.64. 



a — c cos B b= c sin B. 

log c = 8.4359 log c = 8.4359 

log cos B = 9.9276 log sin B = 9.7262 

log a = 8.3635 log 6 = 8.1621 

.-. a = .02309. .-. b = .01452. 



8. cos A — — a— b tan A. 
c 

log 6 = 0.4604 log b = 0.4604 

log c = 0.7084 log tan A = 0.1645 

log cos A = 9.7520 log a = 0.6249 

.-. u4=55°36.1'. .-. a = 4.216. 



9. a = 6tanJ.. 



cos^. 

log 6 = 3.6281 log b = 3.6281 

log tan A = 0.1179 log cos A = 9.7826 

log a = 3.7460 log c = 3.8455 

.-. a = 5571. .-. c = 7007. 



10. tan A = -- 

b 

log a = 2.0043 
log b = 2.0645 

log tan A = 9.9398 log c = 2.1870 

.-. ^1 = 41° 2.4'. /. c= 153.8. 



11. 



c = 


a 


sin A 


loga = 


. 2.0043 


sin .4 = 


: 9.8173 



sin A 
log a = 2.1995 
log sin A = 9.8368 

log b = 2.2243 log c = 2.3627 

.-. b = 167.6. .-. c = 230.5. 



CHAPTER TIL — PAGE 70. 35 

12. a = c sin J.. 6 = c cos A. 

log c= 1.5531 logc=1.5531 

log sin .4 = 9.9314 log cos A = 9.7 102 

log a = 1 .4845 log b = 1 .2693 

/. a = 30.51. /. b = 18.59. 



13. 



14. 



c tan J. 

log a = 2.3100 log a = 2.3100 

log c = 2.4398 log tan A = 0.0436 

log sin A = 9.8702 log b = 2.2664 

.-. .4 = 47° 52.5'. .% 6 = 184.7. 



c = 


b 
sin i? 


log 6 = 


: 0.2158 


sin B = 


: 9.6990 



tan B 
log b = 0.2158 

log tan .8=9.7614 

log a = 0.4544 log c = 0.5168 

.-. a = 2.847. .-. c = 3.287. 



15. a = bta.il A. 



cos A 

log6= 1.1220 log b =1.1220 

log tan A = 9.6114 log cos A = 9.9665 

log a = 0.7334 log c = 1.1555 

.-. a = 5.4125. .-. c = 14.306. 



16. a = c cos B. b = c sin i?. 

log c = 9.8611 log c = 9.8611 

log cos # = 9.9922 log sin B = 9.2747 

log a = 9.8533 log 6 = 9.1 358 

.\ a =.7133. .'. b = .1367. 



17. tan A-- C 



b sin A 

log a = 2.8051 log a = 2.8051 

log b = 2.7000 log sin A = 9.8957 

log tan .4 = 0.1051 log c = 2.9094 

.-. A = 51° 51.9'. .\ c = 811.7. 



36 KEY TO ESSENTIALS OF TRIGONOMETRY. 
18. 



6- 


a 




tan A 


loga = 


: 2.3092 


log tan A — 


: 0.6832 



20. b = a tan B. 



2L tan^L = a 



sin A 

log a = 2.3092 
log sin A = 9.9908 

log b =1.6260 log c = 2.3184 

.-. 6 = 42.27. /. c = 208.15. 



19. co&A = - a=bt£LnA. 

c 

log b = 8.3974 log b = 8.3974 

log c = 8.6805 log tan A = 0.2143 

log cos .4 = 9.7169 log a = 8.6117 

.\ A = 58° 35.7'. .-. a =.0409 



cos B 

log a = 3.2731 log a = 3.2731 

log tan B = 8.6085 log cos B = 9. 



log 6 =1.8816 log c = 3.2735 

,% 6 = 76.13. ,\ c = 1877. 



6 sin A 

log a = 1.3922 log a = 1.3922 

log b = 1.5188 log sin A = 9.7771 

log tan A = 9.8734 log c = 1 .6151 

.-. ^1 = 36° 45.9'. ,\ c = 41.22. 

22. cosJ. = - a=6tan^. 

c 

log 6 = 0.1574 log 6 = 0.1574 

log c = 0.5397 log tan A = 0.3413 

log cos A = 9.6177 loga = 0.4987 

.-: A = 65° 30'. .-. a = 3.153. 

23. a = c cos 1?. 6 = c sin 5. 

log c = 4.5706 log c = 4.5706 

log cos B= 9.9975 log sin B = 9.0337 

log a = 4.5681 log b = 3.6043 

.-. a = 36992. .-. 6 = 4021. 



CHAPTER VII. — PAGE 70. 37 



24. a = b tan A. 



25. 



26. 



cos A 

log b = 2.3011 log 6 = 2.3011 

log tan A = 0.3122 log cos ^4 = 9.6415 

log a = 2.6133 log c = 2.6596 

/. a = 410.5. /. c = 456.7. 



sin A 

log a - 2.5316 log a = 2.5316 

log 6 = 2.3649 log sin A = 9.9172 

log tan A = 0.1667 log c = 2.6144 

/. .4=55° 44.1'. .-. c = 411.5. 



c tan ^4 

log a = 0.2327 log a = 0.2327 

log c = 0.3012 log tan A = 0.2155 

log sin A = 9.9315 log b = 0.0172 

.-. ^4=58° 40'. /. 6=1.0405. 



27. 6 = a tan 5. 



cos B 

log a = 9.9144 log a = 9.9144 

log tan B = 9.5968 log cos B = 9.9685 

log b = 9.51 12 log c = 9.9459 

.-. 6 = .3245. .-. c = .8828. 



28. a = csinA. b^ccosA 

log c = 2.4403 log c = 2.4403 

log sin A = 9.9828 log cos A = 9.4402 

loga = 2.4231 log b =1.8805 

.-. a = 264.9. .-. 6=75.95. 



29. 



tan B sin B 

log b = 2.0800 log b = 2.0800 

log tan B = 9.8329 log sin B = 9.7503 

log a = 2.2471 log c = 2.3297 

/. a = 176.64. .-. c = 213.65. 



38 KEY TO ESSENTIALS OF TRIGONOMETRY. 

30. 



b sin A 

log a = 1 .0046 log a = 1 .0046 

log b = 1.2381 log sin A = 9.7027 

log tan A = 9.7665 log c = 1.3019 

.-. .4 =30° 17.2'. .-. c = 20.04. 

31. c = 2£cos^l. QA , ic 

34. cos A — — • 

log2b= 1.8462 « 

log cos J. = 9.5553 log | c = 1.7268 

7^77 "log a =1.8989 

log c= 1.4015 s 

.-. c = 25.206. lo g cos A = 9.8279 

.-. .4 = 47° 42.9', 

32. a = -i^-. 35. a = -i£_. 

cos 5 cos ^4 

log J c = 0.1886 log i c = 9.4489 

log cos J5 = 9.9493 log cos A = 9.5274 

log a = 0.2393 log a = 9.9215 

.-. a = 1.735. .-. a = .8346. 



36. 



'*« 



33. c = 2b sin J a sin \ C 

log 2 b = 3.6239 log J c = 1.6788 

log sin J C = 9.8116 log sin J 0= 9.9864 

log c = 3.4355 log a = 1.6924 

.-. c = 2725.6. .-. a = 49.25. 

37. Let O be the centre, and AB any side of the pentagon. 
Join OA, and draw OC perpendicular to AB. 

Then AB = 2 OA sin .4 C = 24 sin 36°. 

log 24 =1.3802 

log sin 36° = 9.7692 

log AB= 1.1494 

.-. AB = 14.106. 

38. Let ^4 be the point of observation, B the top of the tower, and 
C its base. 

Then B C = A C tan .4 = 100 tan 38°. 

log 100 = 2.0000 

log tan 38° = 9.8928 

log #0=1.8928 

.-. BC= 78.12. 



CHAPTER VII. — PAGE 71. 89 

39. Let B be the top and G the base of the tower, and A the extrem- 
ity of its shadow. 

m + a BG 103.7 

Then tan .4 = = 

AG 167.3 

log 103.7 = 2.0157 

log 167.3 = 2.2235 

log tan 4 = 9.7922 
.-. A =31° 47.1'. 

40. Let AB be the chord, and the centre of the circle. 
Join OA and OB, and draw OG perpendicular to AB. 

Then smA0G=^=^^- 

OA 1634 

log 513.5 = 2.7105 
- log 1634 =3.2132 

log sin AO G =9.4973 
.-. 40C= 18° 18.95'. 
.-. AOB = 36° 37.9'. 

41. Let A be the top of the mountain, B the remotest point visible, 
and the centre of the earth. 

Then in the right triangle OAB, OA = 3956 + 1J = 3957.25, and 
OB = 3956. 
Hence AB = V OA 2 - OS 2 



= V(04 + OB)(OA - OB) 

= V7913.25 X 1.25. 
log 7913.25 = 3.8984 
log 1.25 = 0.0969 

2 )3.9953 

log .45=1.9976 

.-. AB = 99.45. 

42. Let AB and BGhe consecutive sides of the pentagon. 
Join AG, and draw BD perpendicular to AG 
Then AG = 2 AB cos BAG = 14.056 cos 36°. 

log 14.056 =1.1478 
log cos 36° = 9.9080 

log 4(7=1.0558 

.\ 4C=11.371. 



40 KEY TO ESSENTIALS OF TRIGONOMETRY. 

43. Let A denote the angle of elevation. 

Then . tan4=— • 

060 

log 238 = 2.3766 
log 660 = 2.8195 

log tan A = 9.5571 
.-. A = 19° 50'. 

44. Let A be the position of the buoy, B the top of the light-house, 
and C its base. 

Then AC =BC cot BAC = 133 cot 18° 25>. 

log 133 = 2.1239 
log cot 18° 25' = 0.4776 

log AG= 2.6015 
.\ AC =399.5. 

45. Let H be the position of the headland, S the first point of obser- 
vation, and S' the second. 

Then in the right triangle HSS', SS f = 16.38, and Z SHS' = 33°. 
Hence HS= 16.38 cot 33°, 

and HS f =l^-. 

sin 33° 

log 16.38 =1.2143 
log cot 33° = 0.1875 

logoff =1.4018 
.-. HS= 25.22. 

log 16.38 = 1.2143 
log sin 33° = 9.7361 

log HS f = 1.4782 
.-. HS' = 30.07. 

46. Let AB be the chord, and O the centre of the circle. 
Join OA, and draw OC perpendicular to AB. 

T , . nA AC 20.68 

Then ° A =^AOC = sin 720 48.5'* 

log 20.68 = 1.3156 
log sin 72° 48.5' = 9.9801 

log OA = 1.3355 
,\ 04 = 21.65. 



CHAPTER VII. — PAGE 72. 41 

47. Let be the centre, and AB any side of the octagon. 
Join OA, and draw 00 perpendicular to AB. 

Then OC=^Ccot^OC=6cot22°30', 

d OA = AC - 6 

sin AOC sin 22° 30'' 

log 6 = 0.7782 

log cot 22° 30' = 0.3828 

log 0(7= 1.1610 

.-. 0C= 14.487. 

log 6 = 0.7782 

log sin 22° 30'' = 9.5828 

log OA = 1.1954 
.♦. 0^1=15.682. 

48. Let A be the position of the observer, B the top of the pole, and 
C its foot. 

Then A C = B O cot BA O = 80 cot 10°. 

log 80 =1.9031 
log cot 10° = 0.7537 

log AO= 2.6568 
.-. .40=453.7. 

49. Let be the centre, and AB any diagonal of the pentagon. 
Join OA, and draw O perpendicular to AB. 

Then OA= AC = "ilL 

sin ^00 sin 72° 

log 16.415 =1.2152 

log sin 72° = 9.9782 

log 0-4= 1.2370 
.-. 0-4 = 17.26. 

50. Let I? be the top, and C the foot of the tower, and A the ex- 
tremity of the base line. 

Then BO= AOtsrn BAC = 1000 tan 21° 16' 37". 

log 1000 = 3.0000 
log tan 21° 16' 37" = 9.5904 

log BC= 2.5904 
.-. £0=389.4, 



42 KEY TO ESSENTIALS OF TRIGONOMETRY. 

51. Let AB be the chord, and the centre of the circle. 
Join OA, and draw OC perpendicular to AB. 

Then AB = 2 OA sin AOC = 1446.58 sin 17° 36.5'. 

log 1446.58 = 3.1604 
log sin 17° 36.5' = 9.4807 

log AB =2.6411 
.-. .45 = 437.6. 

52. Let be the centre, and AB any side of the hexagon. 
Join OA, and draw OC perpendicular to AB. 

Then AB = 2 C tan A C = 10 tan 30°. 

log 10 =1.0000 
log tan 30° = 9.7614 

log AB = 0.7614 
.-. .4.3=5.773. 

53. Let A be the position of the first boat, and B of the second ; let 
G be the top of the light-house, and D its foot. 

Then AD = CD cot CAD = 200 cot 14°, 

and BD = CD cot CBD = 200 cot 32°. 

log 200 = 2.3010 
log cot 14° = 0.6032 

log AD= 2.9042 
/. AD = 802. 

log 200 = 2.3010 
log cot 32° = 0.2042 

log BD = 2.5052 
.-. BD= 320.1. 
.-. AB =AD-BD = 481.9. 

54. Let ^1 be the position of the light-house, and B, C, and _D, the 
positions of the ship at 7 a.m., 7.30 a.m., and 10 a.m., respectively. 

Then BC= AB tan BAC = 10.32 tan 18° IS'. 

log 10.32 = 1.0136 
log tan 18° 13' = 9.5174 

log 5(7=0.5310 
.-. BC= 3.396. 
Therefore the rate of the ship is 2 X 3.396., or 6.792 miles an hour. 



CHAPTER VII. — PAGES 72, 74. 43 

a • «. 7, < r> BD 20.376 

Again, tan BAD = = 

AB 10.32 

log 20.376 =1.3091 

log 10.32 = 1.0130 

log tan BAD = 0.2955 

.-. BAD=63°8A'. 

Therefore the bearing of the light-house at 10 a.m. is 63° 8.4' west 

of north. 

Art. 112. — Page 74. 

2. 2AT=a 2 cotA 6. 47?=c 2 sin2A 
2 log a = 2.6916 2 log c = 4.5708 

log cot A = 0.4485 log sin 2 A = 9.9455 

log 2 JT= 3.1401 log 4 AT = 4.5163 

2K= 1380.6 4JT=32831 

.-. AT =690.3. .-. K =8208. 

3. 2AT=a 2 tanjB. 7. 2 7T=6 2 tanA 
2 log a = 9.8290 2 log b = 7.4332 

log tan 5 = 9.6510 log tan A = 0.2190 

log 2A r = 9.4800 log 2 A^ = 7 .6522 

2 AT= .302 2 JT- .00449 

/. AT= .151. ... AT= .002245. 

4. 2K=ab. g 2I = aV(c + a)(c-c). 
log a = 2.1741 log a = 9.9694 

log 6 = 1.8824 i log ( c + a ) = 0.2851 

log 2 K= 4.0565 i log ( c _ a ) = Q.1341 

2iT=11389 log 2 iT= 0^886 

.-. AT=5695. 2 JT=2.447 

.-. K= 1.223. 



5. 2iT=&V(c + &)(c-&). 

log 6 = 9.4851 9. 4 iT=c 2 sin 2^. 

J log (c + 6) = 9.9924 2 log c = 2.8718 

J log (c - 6) = 9.7748 log sin 2 B = 9.7604 

log 2 AT = 9.2523 log 4 AT = 2.6322 

2 iT=. 17876 4 AT =428.7 

.". AT= .08938. .-. AT =107.2. 

10. 2K = b 2 cot B. 

2 log b= 9.0574 

log cot 5 = 0.2508 

log 2 K= 9.3082 

2A'= .2033 

.-. K= .1017. 



44 KEY TO ESSENTIALS OF TRIGONOMETRY. 



CHAPTER IX. 

Art. 121. — Page 84. 



b = a sin £ esc 


A. 


e = a sin Ccsc-4. 


log a = 1.0000 




log a =1.0000 


log sin £ = 9.9890 




log sin (7=9.9567 


log esc A = 0.2107 




log esc 4 = 0.2107 


log b =1.1997 


logc =1.1674 


.-. 6=15.837. 




.-. c = 14.703. 


3. A = 180° - 154° = 26 c 


» 




a — b sin A esc 


B. 


c = 6 sin (7 esc B. 


log b = 9.9051 




log 6 = 9.9051 


log sin A = 9.6418 




log sin C= 9.9909 


log esc B = 0.1015 




log esc £ = 0.1015 


log a = 9.6484 


logc = 9.9975 


.*. a = .445. 




.-. c = .9942. 


4. C = 180° -80° 35' = 99° 25'. 




a = c sin A esc 


C. 


b — c sin B esc C. 


logc == 8.5051 




logc = 8.5051 


log sin A = 9.7706 




log sin £ = 9.8453 


log esc C = 0.0059 




log esc C= 0.0059 


log a = 8.2816 


log b = 8.3563 


.-. a =.01913. 




.-. b = .02272. 


5. B = 180° - 120° 55' = 


59° 5'. 




a = b sin A esc 


B. 


c = 6 sin C esc B. 


log 6 =1.4625 




log 6 =1.4625 


- log sin ^1=9.9996 




log sin C= 9.7390 


log esc £=0.0666 




log cse £ = 0.0666 


log a = 1.5287 




logc = 1.2681 


.-. a =33.78. 




.-. c= 18.54. 



CHAPTER IX. — PAGE 84. 

6. 4 =180° -139° 23' = 40° 37'. 

b — a sin B esc A. c = asin CcscA. 

log a = 0.7340 log a = 0.7340 

log sin £ = 9.9954 log sin C= 9.8170 

log esc .4 = 0.1865 log esc .4 = 0.1805 



log b = 0.9159 






logc = 0.73 1 5 


.-. b = 8.24. 






.-. c = 5.464. 


7. J3=1S0 C -158 C 54' = 


21 c 6'. 






a = c sin A esc 


C. 




5 = c sin I? esc (7. 


logc = 8.2068 






logc = 8.2068 


log sin J. = 9.7613 




log 


sin B =9.5563 


log esc C= 0.0796 




log 


cse C= 0.0796 


loga = 8.0477 


log 6= 7.8427 


.\ a = .011162. 






.-. b = .006962. 



j5 - i80 c - 114 c 28' = 65° 32'. 

b — a sin B esc A. c = a sin (7 esc A. 

log a = 2,6021 log« = 2.6021 

log sin B = 9.9591 log sin C = 9.9375 

log cse .4 = 0.0895 log cse A = 0.0895 



9. 



log b = 2.6507 






logc = 2.6291 


.-. 6 = 447.4. 






.-. c = 425.7. 


£ = 180° — 125° 13' = 


54 c 


! 47'. 




a = 6 sin A esc 


:B. 




c = b sin CcscB, 


log 6 = 2.4973 






log 6 = 2.4973 


log sin A = 9.9652 






log sin C= 9.9122 


log esc £ = 0.0723 






log esc £=0.0723 


log a = 2.5348 


logc = 2.4818 


.-. a = 342.6. 






.-. c = 303.3. 



10. .4 = 180 c - 75 c 28' 18" = 104° 31' 42 / '. 

a = c sin ^4 esc C. 6 = c sin J3 esc C. 

log c = 0.8954 log c = 0.8954 

log sin A = 9.9858 log sin B = 9.7248 

log esc C = 0.1628 log esc C = 0.1628 

log a = 1.0440 log 6 = 0.7830 

.-. a = 11.067. .. 6 = 6.067. 



46 KEY TO ESSENTIALS OF TRIGONOMETRY. 



Art. 122. — Page 85. 

2. tanUi- C) = ^L£ tan i ( J. + C). 6 = asin .BcscA 

a + c 

a-c=12 log =1.0792 loga = 1.4314 

a + c = 42 colog = 8.3768 log sin 5 = 9.8569 

J (A + O) = 67° log tan = 0.3721 log esc A = 0.0080 

log tan ±(A- C) = 9.8281 log b = 1.2963 

... i (4 _ C) = 33° 56.7'. .-. 6 = 19.78. 

.-. ^ = } (^ + C) + i(J- C) = 100° 56.7', • 
and G= i(A + C) - J (4- <7) = 33° 3.3'. 

3. tan b(A — B) = ^— ^ tan * (^4 + J3). c = a sin Ccsc A 

a + b 

a-b=m log = 2.1430 log a = 2.6866 

a+b = 833 colog- = 7.0794 log sin C = 9.8941 

%(A + B) = 64° 12' log tan = 0.3157 log esc A = 0.0030 

log tan I {A — B) = 9.5381 log c = 2.5837 

.-. £ (^ - B) = 19° 2.7'. .-. c = 383.5. 

/. A = i (A + E) + | (,4 - 5 ) = 83° 14.7', 
and .B = j- (J. + 5) - J (J. - 5) = 45° 9.3'. 

4. ttrni (C - B) = c -^t&ni (C + B). a = bsmAcscB. 

c + b 

c — b = 1.265 log - 0.1021 log 6 = 0.3621 

c + 5= 5.869 colog =9.2315 log sin A= 9.9459 

J ( O + 5) = 59° log tan = 0.2212 log esc B = 0.1987 

log tan i (C - B) = 9.5548 log a = 0.5067 

... i ( e - 5) = 19° 44.2'. .-. a = 3.211. 

.\ C=i(C+B) + i(C-B) = 78°U:2 r , 
and 5 = J (0+ J5) - |(0- JB) = 39° 15.8'. 

5. tan i(B- A) = h -^ tan h(B+A). c = a sin Ccsc A. 

b + a 

6_a=.063 log = 8.7993 log a = 9.4771 

b + a = .663 colog = 0.1785 log sin C = 9.9137 

i(B+A) = 27° 32' log tan = 9.7171 log esc A = 0.3790 

log tan l(B — A) = 8.6949 log c = 9.7698 

/. j (J3 - .4) = 2° 50.2'. .\ c = .5886. 

... B=i(B+A) + i(B-A) =30° 22.2', 
and ,4 = J (5 + A) - J (Z? - ^4) = 24° 41.8'. 



CHAPTER IX. — PAGE 85. 47 

6. tan. 1 , (5— C) = ^^tan. 1 , (B + C). a = b sin ^1 esc B. 

b + c 

6_c = 835.8 log = 2.9221 log 6 = 3.0763 

6 + c = 1548.4 colog = 6.8101 log sin ^ = 9.6460 

i(B+ C) = 76° 52' log tan = 0.6320 log esc B = 0.2254 

log tan J (B- C) = 0.3642 log a = 2.9477 

... J (J? - C) = 66° 37.1'. .-. a = 886.6. 

.-. B=i(B+ C) + i(B- C) = 143° 29.1', 
andC=K-B+ 0)— K-B- C) = 10° 14.9'. 



7. tan^(C-^) = - -tan*(C+^). 6 = asin.BcscA 

c -\- a 

c-a= 4.039 log = 0.6063 log a = 0.8692 

c + a = 18.839 colog = 8.7249 log sin B = 9.9962 

i ( O + A) = 48° 47' log tan = 0.0575 log esc J[ = 0.2411 

log tanJ(C— A) = 9.3887 log 6 = 1.1065 

... J (C - A) = 13° 45.1'. .-. b = 12.78. 
.-. C=i(C+^) + J(C-^)=62°32.1', 

and^=K^+^)-K c '-^) = 350 1-9'- 



8. tan}(^l — B) = - — -tan$(^ + B). c = a sin (7 esc A 

a + 6 

a — 6=11.66 log =1.0667 log a =1.7265 

a + b = 94.88 colog = 8.0228 log sin C = 9.9913 

i(A+B) = 50° 43.5' log tan = 0.0874 log esc A = 0.0657 

log tan i(A-B) = 9.1769 log c = 1.7835 

... ±_(A-B)= 8 C 32.8'. .-. c = 60.74. 

.-. 4 = } (4 + #) + } (A -B) = 59° 16.3', 
and B = J (A + 5) - £ {A - B^ = 42° 10.7'. 



9. tanJ(C-^) = - — -tan*((7+£). a = bsmAc8cB. 

c + 6 

c-6=. 02424 log = 8.3845 log 6 = 8.4262 

c + 6 = .0776 colog = 1.1101 log sin A = 9.9545 

J( (7 + B) = 32° 6.5' log tan = 9.7976 log esc B = 0.4453 

log tan J (C— J5) = 9.2922 log a = 8.8260 

... i ( c - 5) = 11° 5.3'. .-. a = .06699. 
.-. C=i(C+B) + ilc-B) = 43°11.8 f , 
and B = J (C-f £) - |(C- 5) = 21° 1.2'. 



48 KEY TO ESSENTIALS OF TRIGONOMETRY. 

10. taiii(C-A) = c -^tani(C+A). b = asinBescA. 
c + a 

c- a = 16.66 log =1.2191 loga = 1.7108 

c + a= 119.32 colog = 7.9233 log sin B = 9.9923 

i(C+^L) = 50° 23' 43" log tan = 0.0823 log esc A = 0.1842 

log tan i (C- A) = 9.2247 log 6 = 1.8873 

... J(a-^)=9°31.4 / = 9°31'24". /. 6 = 77.14. 
... C=i(C+A) + i(C-A) = b9°55 f V f , 
and A = J (<7 + 4) - J (O- A) = 40° 52' 19". 



Art. 123. — Page 88. 

3. Here s = 4.5, s-^a = 2.5, s — 6=1.5, s — c = .5. 

log ( s _ a) = 0.3979 log r = 9.8099 

log ( s - 6) = 0.1761 log (s - b) = 0.1761 

log - e) = 9.6990 log tftn j B = g^ 

colog s = 9.3468 i ^ = 23° 17 1'. 

2)9.6198 .-. J3 = 46°34.2'. 

log r = 9.8099 logr = 



log r = 9.8099 log (s - c) = 9.6990 

log (*-<*) = 0.3979 logtanJC=<U109 

log tan i A = 9.4120 i C = 52° 14.2'. 

J .4 = 14° 28.7'. .-. C= 104° 28.4'. 

.-. ^4 =28° 57.4'. 

Check, ^ + 5+ (7=180°. 



4. Here s = 8.5, s — a = 4.5, s — b= 1.5, s — c = 2.5. 

log (s-a) = 0.6532 log r = 0.1489 

log ( s - 6) = 0.1761 log (s - b) = 0.1761 

log ( S - C ) = 0.3979 lQg tan i B = ^7 8 

colog s = 9.0706 i 5= 43 o 12 .4'. 

2 )0.2978 .-. JB=86°24.b'. 

log r = 0.1489 l og r = 0.1489 

log (s - a) = 0.6532 log (s - c) = 0.3979 

log tan J J. = 9.4957 log tan \ = 9.7510 

} .4 =17° 23.2'. iC=29°24.5'. 

.-. ^4 = 34° 46.4'. .v 0=58° 49'. 

Check, ^L + £ + C= 180° 0.2 . 



CHAPTER IX. — PAGE 88. 49 

5. Here s = 7.4, s — a = 1.8, s — b = 3.1, 8 — c = 2.5, 

log (* — a) = 0.2553 log r = 0.1377 

log ( S - 6) = 0.4914 log (s - b) = 0.4914 

log (s - c) = 0.3979 lQg tRn j fl = 96463 

colog 5 = 9.1308 J £ = 23= 53.2'. 

2)0.2754 .-. £=47° 46.4'. 

log r = 0.1377 log;— 0.1377 

log (s - «> = 0.2553 log ( s _ c ) = 0.3979 

log tan J .4 = 9.8824 log tan J C = 9.7398 

M= 3 " c 20'. £C = 28° 46.7'. 

••• ^ = 74° 40'. ... C=57°33.4'. 

Check, ^1 + ^ + C = 179 c 59.8'. 

6. Here s = .344, s — a = .114, s — 6 = .084, s - c = .146. 

log (s - a) = 9.0569 log r = 8.8045 

log (s - 6) = 8.9243 log (s -b) = 8.9243 

log ( S - c) = 9.1644 tan } ^ = ^^ 

colog s = 0MU J 2? = 37° 11.9'. 

2 )7.6090 .-. £=74-23.8'. 

log r = 8.8045 log r _ 8i8 045 

log (* — a) = 9.0569 log ( s _ c ) = 9.1644 

log tan J JL = 9.7476 log tan i<7 = 9.6401 

M = 29°13'. |C=23<>35.3'. 

.-. ^==58°26'. ... c = 47° 10.6'. 
Check, .4+1?+ C'=180 c 0,4'. 

7. Here s = 120.2, s — a = 40.9, s - b = 26, s - c = 53.3. 

log (s - a) = 1.6117 logr = 1.3367 

log (s-b) = 1.4150 log (s-b)= 1.4150 

log (,- = 1.7267 logtani£ = ^I7 

• colog s=_^9201 1 2? =39° 51.9'. 

2)2.6735 .-. £=79° 43.8'. 

log r = 1.3367 log r = 1 3357 

log (s-a)= 1.6117 log ( 5 - c) = 1.7267 

log tan J J. = 9.7250 log tan 1 c= 9-6100 

I ^4 = 27° 57.7'. £(7= 22° 10'. 

.". J. = 55° 55.4'. . C=44°20'. 

Check, .4 + 5+0= 179° 59.2'. 



50 KEY TO ESSENTIALS OF TRIGONOMETRY. 

8. Here s = 542, s-a = 221, s — b = 181, 5 - c = 140. 

log (s - a) = 2.3444 log r = 2.0071 

log (s - 6) = 2.2577 log (s - o) = 2.2577 

log (s-c) = 2.1461 1 * id n-m. 

1 -o««a 1 log tan J jB= 9./ 494 

eologs = ^2660 j iB=29Cl9>. 

2 )4.0142 .-. £=58° 38'. 

log r = 2.0071 log r = 2.0071 

log (s - a) = 2.3444 log ( s - c) = 2.1461 

log tan M = 9.6627 log tan*C = 9.8610 

J .4 = 24° 42.1'. 1 C = 35° 58.9'. 

.-.^1 = 49° 24.2'. ... C = 71° 57.8'. 

Check, ^L+# + C=180°. 

9. Here s = .936, s — a=± .295, s — b = .407, s - c = .234. 

log (s — a) = 9.4698 log r = 9.2386 

log (s — 6) = 9.6096 log (s-b) = 9.6096 

log (,-c) = 9.3692 log tan* 2* =5^5 

eologs = 00287 J 2? = 23° 3.1'. 

2 )8.4773 /.B = 46° 6.2'. 

log r= 9.2386 log r = 9.2386 

log (s - a) = 9.4698 log ( s - c ) = 9.3692 

log tan I A = 9.7688 log tan 1 c= 9 8694 

M= 30° 25.4'. J C= 36° 30.8'. 

.-. .4= 60° 50.8'. ... c=73°1.6'. 

Check, A + B + C= 179° 58.6'. 

10. Here s = 6.989, s-a = 3.97, s-b= .258, s-c = 2.761. 
log (s — d) = 0.5988 log r = 9.8035 

log (s — 6) = 9.4116 log (s — 6) = 9.4116 

log (.-*) = 0.4411 log tan *5 = 03919 

cologs = 9.1556 JjB= 67 o 55 ^ 

2 )9.6071 .-. B= 135° 50.6'. 

log r = 9.8035 l og r = 9.8035 

log (s - a) = 0.5988 l og (s - c) = 0.4411 

log tan J A = 9.2047 l og tan 1 q= 9.3624 

M= 9° 6.2'. 1 C= i 2 o 68.3'. 

.-. .4=18° 12.4'. ... 0=25° 56.6'. 
Check, A+B+ (7=179° 59.6'. 



CHAPTER IX. — PAGES 91, 92. 51 



Art. 127. — Pages 91 and 92. 

6. Since b is < a, there is but one solution, corresponding to the acute 
value of B. 

• t> b sin A • n A 

sin_B = c = asm G esc A. 

a 

log b = 0.5551 log a = 0.7059 

colog a = 9.2941 log sin C = 9.9884 

log sin A = 9.9530 log esc A = 0.0470 

log sin £ = 9.8022 log c = 0.7413 

.-. B = 39° 21.3', .-. c = 5.511. 

and 0= 180° - 103° 11.3' = 76° 48.7'. 



7. Since 6 is > c, and O is acute, there will be two solutions, one 
solution, or no solution, according as log sin B is negative, zero, or 
positive. 

sin B = — — — a x =b sin A x esc B. a 2 = b sin A 2 esc B. 

c 

log b = 1.8739 log b = 1.8739 log 6 = 1.8739 

colog c = 8.2062 log sin A x = 9.9408 log sin ^4 2 = 9.0316 

log sin C = 9.6615 log esc B = 0.2584 log esc B = 0.2584 

log sin B = 9.7416 log a x = 2.0731 log a 2 = 1.1639 

/. ^ = 33° 28.4', .-. a x = 118.33. 

and .B 2 = 146° 31.6'. 

... ^ - 180° - 60° 46.4' = 119° 13.6', 
and A, = 180° - 173° 49.6' = 6° 10.4'. 



8. Since c is < 6, there is but one solution, corresponding to the 
acute value of C. 

. /-y c sin B . i • a -r> 

sin C = a = b sin J. esc 5. 

b 

log c = 9.2971 log b = 9.3687 

colog b = 0.6313 log sin A = 9.4825 

log sin B = 9.9757 log esc B = 0.0243 

log sin C = 9.9041 log a = 8.8755 

.-. 0=53° 18.9', .-. a =.07508. 
and A = 180° - 162° 18.9' = 17° 41.1'. 



52 KEY TO ESSENTIALS OF TRIGONOMETRY. 

9. Since a is < c, there is but one solution, corresponding to the 
acute value of A. 

4 a sin C , . -r, A 

sin A = ■ b = a sin B esc A. 

c 

log a = 0.0294 log a = 0.0294 

colog c = 9.7670 log sin B = 9.8916 

log sin C = 9.7228 log esc A = 0.4808 

log sin A = 9.5192 log b = 0.4018 

.-. A= 19° 18.1', /. & ='2.522. 

and .£ = 180° - 51° 11.1' = 128° 48.9'. 

10. sin A = c l = a sin C x esc A. c 2 = a sin C 2 esc A. 

log a = 9.2704 log a = 9.2704 log a = 9.2704 

colog 6 = 0.7696 log sin C Y = 9.7795 log sin C 2 = 9.4314 

log sin 5 = 9.9524 log esc A - 0.0076 log esc A = 0.0076 

log sin A = 9.9924 log c Y = 9.0575 log c 2 = 8.7094 

/. il 1 = 79 o 20', /. c 1= . 11416. /. c 2 =. 05121 

and ^ 2 =100 o 40'. 

... <7 1= 180° -143° =37°, 
and C 2 = 180° - 164 c 20' = 15° 40'. 

11. Since c is > a, and A is obtuse, the triangle is impossible. 

12. Since b is < c, there is but one solution, corresponding to the 
acute value of B. 

-r, 6sin(7 7 • A -r> 

sin B — a = b sin A esc B. 

c 

log b =1.7016 log 6 = 1.7016 

colog c = 8.1752 log sin A = 9.9232 

log sin C = 9.7340 log esc B = 0.3892 

log sin B = 9.6108 log a = 2.0140 

.-. JB = 24°54', .'. =103.3. 

and A = 180° - 56° 54.4' = 123° 5.6'. 

13. S in(7=^^. • b = atSLnB. 

a 

log c = 1.0000 log a = 0.9373 

colog a = 9.0627 log tan B = 9.7623 

log sin ,1= 9.9373 log6 = 06996 

0.0000 .-. b= 5.007. 
.-. O=90°, 
and B = 90° - 59°57' = 30° 3'. 



CHAPTER IX. — PAGE 92. 53 

14. sin C— «! = b sin A x esc B. a 2 = &sin^l 2 cscB. 

b 

log c = 0.8351 log b = 0.7127 log 6 = 0.7127 

colog b = 9.2873 log sin ^ = 9.9695 log sin A 2 = 9.5939 

log sin 5 = 9.8422 log esc B = 0.1578 log esc J5 = 0.1578 

log sin C = 9.9646 log a Y = 0.8400 log a 2 = 0.4644 

/. Ci = 67°10', .-. a 1 = 6.918. /. a 2 = 2.913. 

and C 2 =112°50'. 

... A x = 180° -111° 13' = 68° 47', 

and A 2 = 180° - 156° 53' = 23° 7'. 

15. Since a is < 6, there is only one solution, corresponding to the 
acute value of A. 

A a sin B • n a 

sin A = c = a sin C esc .A. 

6 

log a = 2.3315 log a = 2.3315 

colog b = 7.5455 log sin C = 9.6825 

log sin J5 = 9.9863 log esc .4 = 0.1367 

log sin A = 9.8633 log c = 2.1507 

.-. ^1 = 46° 53.3', .-. c = 141.48. 

and C= 180° - 151° 13.3' = 28° 46.7'. 

lb. smii= 

c 

log b = 3.4870 

colog c = 6.9126 

log sin (7=9.9179 

log sin 5 = 0.3175 
Since log sin B is positive, the triangle is impossible. 

17. Since c is < a, there is only one solution, corresponding to the 
acute value of C. 

sin C = . b = a sin B esc ^4. 

a 

log c = 9.7086 log a = 9.8511 

colog a = 0.1489 log sin B = 9.9363 

log sin A = 9.7606 log esc A = 0.2394 

log sin C = 9.6181 log b = 0.0268 

.-. 0=24° 31.4', ■-. ft =1.0637. 

and B = 180° - 59° 42.4' = 120° 17.6'. 



54 KEY TO ESSENTIALS OF TRIGONOMETRY. 

18. . sinJ?=^A c = 6sin<7. 
a 

log b = 2.2206 log b = 2.2206 

colog a = 7.9712 log sin C = 9.8843 

log sin ,1=^8082 logc=^9 

log sin B = 0.0000 ,\ c = 127.32. 

/. £ = 90°, 

and 0= 90° - 40°0' 21" = 49° 59' 39". 



19. sinJ. = ^-^ — b x = a sin B Y esc A 6 2 = a sin 2? 2 esc A. 

c 

log a = 9.5073 log a = 9.5073 log a = 9.5073 

colog c = 0.5673 log sin B x = 9.9255 log sin £ 2 = 9.4853 

log sin C = 9.8989 log esc A = 0.0265 log esc ^4 = 0.0265 

log sin J. = 9.9735 log 6 X = 9.4593 log b 2 = 9.0191 

/. A x = 70° 12', .-. 6j = .2879. /. 6 2 = .1045. 

and .4 2 =109°48'. 

.-. B t = 180° - 122° 36' = 57° 24', 

and B 2 = 180° - 162° 12' = 17° 48'. 

20. Since c is < 6, there is only one solution, corresponding to the 
acute value of G. 

. n c sin B 7 • a -c> 

sin C = a = b sin A esc 5. 

log c = 2.7828 log b = 2.9092 

colog 6 = 7.0908 log sin A = 9.4596 

log sin B = 9.9075 log esc B = 0.0925 

log sin C = 9.7811 log a = 2.4613 

.-. = 37° 10', .-. a = 289.3. 

and A = 180° - 163° 15' 20" = 16° 44' 40". 



2. 



Art. 

2 K= ac sin £. 
log a =1.5798 
logc= 1.7868 
log sin B= 9.9670 


128.- 


-Page 93. 

3. Here s = 9, 
5 — a = 4, 

s - b = 2, 
and s — c = 3. 




log2JT= 3.3336 

2 isT =2155.7 

.-. K= 1077.9. 


iT = Vs(s-a)(s-6)(s- 


-0 



CHAFTER IX. — PAGE 93. 



55 



logs = 0.9542 
log (s- a) = 0.6021 
log (* - 6) = 0.3010 
log (s - c) = 0.4771 

2 )2.3344 
log#= 1.1672 
.-. # = 14.697. 



4. 2 K = b 2 sin C sin A esc B. 

C= 180 o -106°23'=73°37'. 

2 log b = 0.6320 
log sin C = 9.9820 
log sin A = 9.9730 
log esc £ = 0.2268 

log 2 #=0.8138 
2 #=6.513 
.". # = 3.257. 



5. 2# = 6csinA 

log 6 = 2.0649 
log c = 2.0000 
log sin A = 9.9449 

log 2 # = 4.0098 
2 #=10229 
.-. #= 5114. 



6. Here 



120, 
s — a = 41, 



7. 2 7f= a 2 sin £ sin C esc A 

O=180° -67° 8' =112° 52'. 

2 log a = 0.9892 
log sin J? =9.3822 
log sin C =9.9645 
log esc ,4 = 0.0966 

log 2 # = 0.4325 
2# = 2.707 
.-. #=1.353. 

8. 2#=6 2 sinCsin.4csc£. 

B = 180° - 117°13'=62°47'. 

2 log 6 = 9.2850 

log sin (7 =9.8132 

log sin A = 9.9880 

log esc B = 0.0510 

log 2 #=9.1372 
2#=. 13716 

.-. #=.06853. 

9. Here s = 34, 

5 - a = 10.9, 
s - b = 14.3, 

and s — c = 8.8. 

# = Vs(s-a)(s-6)(s-c). 

log s=1.5315 

log - a) = 1.0374 

log ( s -b) = 1.1553 

log (s-c) = 0.9445 

2)4.6687 



s - 6 = 26, 






log #=2.3343 


and s — c = 53. 






/. #=215.9. 


#= Vs(s — a)(s — &)(«- 


-c). 




log s = 2.0792 




10. 


2#=acsin£. 


log (s- a) = 1.6128 






log a = 9.5089 


log (s- 6) = 1.4150 






log c = 9.9582 


log (s - c) = 1.7243 






log sin £ = 9.9387 



2 )6.8313 
log #=3.4156 
/. #=2604. 



log 2 #=9.4058 
2 #=.2546 
/. #=.1273. 



56 KEY TO ESSENTIALS OF TRIGONOMETRY. 

11. 2iT=c 2 sin^4sin.BcscC. 13. Here s=8.04, 

A = 180° - 131°49'=48°11'. s-a = 2.22, 

2 log c = 3.8088 s-b = 2.04, 

log sin A = 9.8723 and s-c = 3.78. 

log sin B = 9.9932 ^ = Vs(s-a)(s-6)(s-c). 

logcsc 0=0^1 logs = 0.9053 

log 2 iT = 3.9534 log (s - a) = 0.3464 

2 K = 8982 log (s - 6) = 0.3096 

.-. A: =4491. log (s - c) = 0.5775 

12. 2K = ab sin C. 2)2.1388 
log a = 8.0072 log K = 1 .0694 
log 6 = 8.2607 .-. iT= 11.732, 

log sin (7-9.2924 

log 2 K= 5.5603 
2 JY= .00003633 
.-. K=. 00001817. 

Art. 129. — Pages 93 and 94. 

1. Let A be the first point of observation, and B the second ; and 
let C be the top of the tower, and D its base. 

T1 AC _ bui ABC 

llien AB~ sin ACB' 

or, AC =AB sin ABC csc ACB 

= 100 sin 35° 16' esc 17° 23'. 
log 100 = 2.0000 
log sin 35° 16' = 9.7615 
log esc 17° 23' = 0.5247 

.-. log AC= 2.2862 
Then CD = A C sin CAD = A C sin 52° 39', 

and AD = AC cos CAD = AC cos 52° 39'. 

log AC = 2.2862 
log sin 52° 39' - 9.9003 

log CD = 2.1865 
/. CD = 153.64. 

log AC= 2.2862 
log cos 52° 39' = 9.7830 

log AD = 2.0692 
/. AD =117.27. 
.-. BD=AB+AD = 217.27. 



CHAPTER IX. — PAGE 93. 57 

2. Denoting the sides opposite the angles A, B, and C of the trian- 
gle ABC by a, 6, and c, respectively, we have 



K — Vs (s — a) (s — b) (s — c) . 
Here s = 351, s — a = 115, s — b = 40, and 5 - c = 196. 

logs = 2.5453 
log (s - a) = 2.0607 
log (s-6) = 1.6021 
log (s - c) = 2.2923 

2 )8.5004 

log K= 4.2502 

.-. JT= 17792, 

Denoting the sides opposite the angles A, C, and D of the triangle 
ACD by a, c, and d, respectively, we have 

K — Vs (s — a) (s — c) (s — d) . 
Here s = 334, s - a = 82, s — c = 229, and s-d = 23. 

log s = 2.5237 
log (s- a) = 1.9138 
log ls-c) = 2.3598 
log (s-d) = 1.3617 

2 )8.1590 
log K= 4.0795 
.-. iT= 12008. 
Therefore area ABCD = 17792 + 12008 = 29800. 

3. Let A be the position of the first post, and B of the second, and 
let C be the top of the bluff, and D its foot. 

AC sin ABC 



Then 



AB smACB 



Whence, CD = AC sin CAD 

= AB sin ^l^Csin CAD 

sin ACB 
= 1000 sin 9° 33' sin 27° 40 1 
sin 18° V 
log 1000 = 3.0000 
log sin 9° 33' = 9.2198 
log sin 27° 40' = 9.6668 
log esc 18° V = 0.5073 

log CD = 2.3939 
.-. CD= 247.7. 



58 KEY TO ESSENTIALS OF TRIGONOMETRY. 

4. Let A be the starting-point, and B and C the positions of the 
first and second yachts, respectively, at the end of 40 minutes. 

Then in the triangle ABC, we have 

AB = 6M, .4(7=5.14, and Z A = 45°. 

tuniCC - B) = AB ~~ AC tanly (C + B). 
2V J AB + AC " V 

AB - A C = 1 .82 log = 0.2601 

AB + u4(7= 12.1 colog = 8.9172 

J (C + B) = 67° 30' log tan = 0.3828 

log tan h (C-B) = 9.5601 
.-. i(C-£) = 19°57.5', 
and B=i(C+B)-i(C-B) = 47° 32.5'. 

J3<7=.40sin AcscB. 
log ^40= 0.7110 
log sin A = 9.8495 
log esc 5 = 0.1321 

log .8(7 =0.6926 
.-. J3C= 4.927. 

5. Let A be the position of the lighthouse, and B and C the first and 
second positions of the ship. 

Then in the triangle ABC, we have 

BC=U, ZB= 105°, andZC=30°. 
Whence, Z A = 180° - 135° = 45°. 

AB= BC sin C esc A. 

log BC= 1.1461 

log sin (7=9.6990 
log esc .A =0.1505 



log^B = 


: 0.9956 








.-. AB = 


:9.9. 








AC = 


: BC sin 


B 


CSC 


A, 


logBC = 


: 1.1461 








log sin B = 


: 9.9849 








logcsCu4 = 


: 0.1505 









log AC = 1.2815 
.-. .4(7=19.122. 



CHAPTER IX. — PAGE 94. 59 

6. AB= BC sin C esc A. 

A = 180° - 158° 23' = 21° 37'. 

log BG= 2.3187 

log sin (7=9.7218 
log esc A = 0.4337 



log .4i? = 2.4742 

.-. AB = 298. 

7. Let A be the point of observation, B the top of the flag-pole, C 
its foot, and D the base of the tower. 

Then AC = smABC 

BC sin BAG 

or, AC =BC sin ABC esc BAC 

= 40 sin 51° V esc 18° 35'. 

log 40 =1.6021 

log sin 51 c 7' = 9.8912 

log esc 18° 35' = 0.4967 

.-. log AC= 1.9900 

AD = AC cos C^1Z> = .4Ccos 20° 18'. 
log J. (7 =1.9900 
log cos 20° 18' = 9.9722 

log AD = 1.9622 
/. JLD = 91.66. 

CD = ^iCsin CAD = AC sin 20° 18'. 
log .AC= 1.9900 
log sin 20° 18' = 9.5402 

log CD= 1.5302 
.-. CD= 33.9. 

8. DC = J3Z>sin BDC esc BCD 

= DZ)sin60 c csc20 o . 
But, £Z> = ^4Z> sin BAD esc .4£Z> 

= 500 sin 60° esc 80°. 

Whence, B C = 500 sin 2 60° esc 20° esc 80°. 

log 500 = 2.6990 

2 log sin 60° = 9.8750 

log esc 20° = 0.4659 

log esc 80° = 0.0066 

log DC =3.0465 
,\ BC= 1113.1. 



60 KEY TO ESSENTIALS OF TRIGONOMETRY. 

9. A C = CD sin AD C esc CAD. 

log 05 = 2.1761 

log sin AD 0=9.6990 

log esc CAD = 0.086 6 

log AC =1.9617 

.-. .40=91.56. 

BC= CD sin BDCcse CBD. 

log CD = 2.1761 

log sin 550=9.9968 

log esc CBD = 0.3430 

log 50 =2.5159 

.-. 50 = 328. 

JBO + -40 
50-^10=236.44 log = 2.3737 

BC+ AC= 419.56 colog= 7.3772 

' } (5.40 + ABO) = 77° 30' log tan = 0.6542 

log tan J (BAC - ABC) = 0.4051 
.-. J(5^10-^150)= 68° 31.4', 
and 5^0=K J5 ^C+^^^) + K^^C-^^C) = 146° 1.4'. 

.45 = 50sin ACB esc BAC. 

log50= 2.5159 
log sin ACB =9.6259 
log esc 5^10= 0.2527 

log AB =2.3945 

.-. .45 = 248. 

10. Denoting the sides opposite the angles A, B, and O of the trian- 
gle .450 by a, 6, and c, respectively, we have 

K=Vs(s-a)(s-b)(s-c). 
Here, s = 87.5, 

s — a = 24.5, 
s - 6 = 12.5, 
and s — c = 50.5. 

logs = 1.9420 
log (s- a) =1.3892 
log ( s - 6) = 1.0969 
log (s-c) = 1.7033 
2 )6.1314 
log iT= 3.0657 
.-. JT= 1163.2. 



CHAPTER IX. — PAGE 94. 61 



logs = 1.9420 

log(s-a)= 1.3892 

colog b =8.1249 

colog c = 8.4318 

2 )9.8879 
log cos I £.4(7=9.9439 
iBAC=2$°3(y 
.-. BAC=bl°. 

Denoting the sides opposite the angles A, B, and D of the triangle 
ABD by a, b, and d, respectively, we have 





cos i BAD =J< s - a l 
\ bd 


Here, 


s = 49.5, 


and 


s _ a — 7.5. 




logs =1.6946 




log (s-a) = 0.8751 




colog 6= 8.6990 




colog d = 8.4318 




2)9.7005 




log cos i BAD= 9.8502 




J£,4Z>=44 54.2' 




.-. 5.4Z)=89 48.4'. 


/. CAD-. 


= JB.4Z>-£^<7=32 48.4'. 



2 area ^ CD = AC • .4Z> • sin CAD. 
h>gAC= 1.8751 
log AD =1.3010 

log sin (Ml) = 9.7339 

log (2 area ^1 CD) = 2.9100 
2 area ACD= 812.8 
.\ area ACD = 406.4. 
.-. area .45(71) = 1163.2 + 406.4 = 1569.6. 



62 KEY TO ESSENTIALS OF TRIGONOMETRY. 



CHAPTER XI. 

Art. 153. — Page 112. 



5 A am u. . — 

sin A = + 

sine - cosb = £2±l. 

log sin a =9.5543 co _^ a 

log sin c =9.8311 , no ^ c 

& log cose =9.8665 

log sin A = 9.7232 l og cos a = 9.9702 

180° - A = 31° 55'. 

.-. ud=148°5'. 

+ 



6. 



~~„ j. c °s B 
cos 6^ 



log cos b = 9A 

.-. b = 38° 2'. 



cosi?=^. ^Aee*. 

tan c 

__ . ^ cos jB 

sin ^4 = 

log tan a =9.5842 cos b 

log tan c =9.9646 log cos B = 9.6196 

log cos B =9.6196 log cos 6 = 9.8963 

/. .£ = 65° 23.2'. log sin A = 9.7233 

cos A — •+ — 

cos a = . ' A , ,-. 

s j n j$ cos c = cot A cot B. 

log cos J. = 9.8050 log cot A = 9.9187 

log sin B = 9.9252 log cot B = 9.8070 

log cos a =9.8798 log cose =9.7257 

.-. a = 40° 41.8'. 180° - c = 57° 52.5'. 



,\ c= 122° 7.5'. 



sin A 

+ Chech. 

log cos .B= 9.7322 

log sin A = 9.8864 cos c = cos a cos b. 

log cos b =9~^8 log cos a =9.8798 

180°- 6 = 45° 29.2'. logcosb = 9 ' 8458 

/. b = 134° 30.8'. log cos c = 9.7256 



CHAPTER XL — PAGE 112. 63 

7 tan A = tanq . cos c = cos a cos &• 

sin6 log cos a =9.9730 



log tan a = 9.5611 



.4= 149° 41'. 



or 152 c 48.4'. 

tan b 



log cos b =9.8935 



log sin b =9.7941 log cose =9.8665 

logtan^=^o^ 180° -c = 420 40'. 

180° - A = 30° 19'. ■"' c ~ 13/ ° 20 '' 



Chech. 



. 73 tan b 

tan 1* = - cos c tan At&nB= 1. 

sin a 

log tan b =9.9006 log cose =9.8665 

log sin a =9.5341 log tan J. = 9.7670 

logtani? = ^5 log tan i?= 0.3665 

.-. B = m° 43.8'. log 1 = 0.0000 

A cos B . sin 6 

sin .4 = sinc = 

cos b sin B 

log cos B= 9.2397 log sin 6 =9.9661 

log cos 6 = 9.5798 . log sin jB = 9.9934 

log sin ^1= 9.6599 , inc = 9>9727 

•'.^ 27° 11.6', .-. c = 69°54', 



or 110° 6'. 

Chech. 



sine 



tan I? 
log tan b =0.3864 
log tan B = 0.7537 

log sin a = ^7 l0 ^ sina = 9 ' 6327 

,. a = 25° 25.2', l0 ^ sinc = 9 ' 9727 

or 154° 34.8'. log sin A = 9.6600 
Ans. l.A= 27° 11.6', a = 25° 25.2', c= 69° 54'. 
2. .4=152° 48.4', a =154° 34.8', c = 110° 6'. 

ft "" j cot B tan a = cos B tan e. 

9. tan .4 = 

cose log cos 5 =9.5736 

+ log tan c =0.8431 

log cot B =9.6064 

log cose =9.1525 logUna =0.4167 

180° - a = 69° 2.4'. 

log tan .4 = 0.4539 a = 110° 57.6'. 

180° -^1=70°37.5'. 
.-. 4 = 109° 22.6'. 



64 KEY TO ESSENTIALS OF TRIGONOMETRY. 

Check. 



sin b — sin B sin c. 
log sin B = 9.9672 
log sine =9.9956 



log sin b =9.9628 
180° -b = 66° 38'. 
,\ b =113° 22'. 



, . tan a 

tern A = 

sin b 

log tan a =0.4167 
log sin b =9.9628 

log tan .4 = 0.4539 



10. cos A = cos a sin B. 

log cos a =9.6856 
log sin B = 9.9203 

log cos A = 9.6059 
.-. .4 = 66° 12.1'. 



- . + - 

tan b — sin a tan B. 

log sin a =9.9418 
log tan B =0.1765 

log tan b =0.1183 
180° -b =52° 42.6'. 
.-. b =127° 17.4'. 



tan c = 



+ 
tana 

cos B 



log tan a = 0.2562 
log cos £ = 9.7438 

log tan c =0.5124 
180° - c = 72° 54.9'. 
.-. c= 107° 5.1'. 



Check. 
cos A = 



tan b 
tano 

log tan b =0.1183 
log tan c =0.5124 



log cos A = 9.6059 



11. 



tan A = 


tan a 
sin b 


log tan a = 


: 0.2683 


log sin b - 


: 9.7675 


log tan A — 


: 0.5008 


.-. A = 


: 72° 28.9', 


tan B = 


tan b^ 




sina 




+ 


log tan b = 


: 9.8586 


log sin a — 


: 9.9446 



log tan B =9.9140 
180°- £=39° 21.9'. 
.-. J5= 140° 38.1'. 



- + - 

cos c = cos a cos b. 

log cos a = 9.6763 
log cos b =9.9089 

log cose =9.5852 
180° -c =67° 22.3'. 
.-. c= 112° 37.7'. 

Check. 

cos c tan A tan B= 1. 

log cose =9.5852 
log tan .4=0.5008 
log tan B = 9.9140 

log 1 = 0.0000 



CHAPTER XI. -PAGE 112. 55 



12. sin B =~L4. 

_ log sin a =9.9072 

logcos^= 9.2324 log 8 in 4 =9.9936 

log cos a = 9.5736 lo £ sin c = 9.9736 

.-. c=70°14', 
or 109° 46'. 



log sin B = 9.6588 
.*. B=21° 7.2', 
or 152° 52.8'. 



-f- Check. 

sin&=i^. 

ta ^ sin i? = $LA 

sine 
log tan a =0.3936 log sin b =9.6325 

log tan .4 = 0.7611 log sine =9.9736 

log sin b = 9.6325 log sin B = 9.6589 

.-. b = 25° 24.4', 
or 154° 35.6'. 

Ans. 1. B= 27° 7.2', 6= 25° 24.4', c=109°46' 
2. ^=152° 52.8', 6= 154° 35.6', c= 70° 14'.' 

13. cos~4- ta * 5 sini? = $L*. 

tanc 
_ log sin b =9.4130 

log tan b = 9.4281 log sin c = 9 f 668 

log tan c = 9.7196 log sin B = 9.7462 

log cos A = 9.7085 "*' B = 33 ° 62 - 6 ' 

180° -.4 = 59° 15.7'. 
.*. A = 120° 44.3'. 

Check. 

co7a=™*5. sin^=^i^. 

cos 6 cos a 

+ log cos A = 9.7085 

log cos c = 9.9473 lo g cos a = 9.9624 

log cos b =9.9849 logsinJ5 = 9J46T 

log cos a =9.9624 
180°- a = 23° 30'. 
.'. a= 156° 30'. 



66 KEY TO ESSENTIALS OF TRIGONOMETRY. 



14. 



cos A cos c = cot A cot -B. 

siuB log cot A = 9.7075 

log cos A = 9.6573 log cot B __ 0.1219 



log sin £ = 9.7801 



log sin a =9.9398 



tan.Br: 


+ 

cot J. 
cose 


log cot A - 

log COS C - 


= 9.4822 
= 9.6183 



log cose =9.8294 



log cos a =9.8772 . c = 47° 32.1'. 
.-. a -41° 5.5'. 

i cos B n i 7 

cos o = Check. 

sin A 

log cos £ = 9.9019 cos c=cosa cos 6. 

log sin Jl =9.9498 log cos a =9.8772 

i i n n-oi log COS 6 =9.9521 

log cos 5 = 9.9o21 & 

.-. 6 = 26 c 25'. log cose =9.8293 



15. sin a = sin A sin c. — + — 

i • a c\ c\of\(\ tan b = cos A tan c. 

log sin J. = 9.9809 

log sin c = 9.9589 lo S cos ^ = 9.4630 



log tan c =0.3406 



a = 60° 31.4'. log tan 6 =9.8036 

180°- b = 32° 27.9'. 
/. b =147° 32.1'. 





Check. 






tSLYlB = 


tan b 
sin a 


log 


tan 6 = 


: 9.8036 


log 


sin a = 


: 9.9398 



log tan B =9.8639 
180° — B = 36° 10'. 

,\ B = 143° 50'. log tan B = 9.8638 



« n . A cos B . tan b 

16. sin ^4= — sma = 

cos b tan jB 

log cos B =9.9129 log tan b =9.8202 

log cos b =9.9213 log tan B= 9.8468 

log sin .4 = 9.9916 log sin a =9.9734 
.-. .4 = 78° 46.7', .-. a=70 c 10', 

or 101 c 13.3'. or 109° 50'. 



CHAPTER XI. -PAGE 112. 67 

sin c= si!Ll. Check. 

sinB sin a 



sin c 



log sin b =9.7415 sin A ~ ' 

log sin B = 9.7597 i og sin a = 99734 

log sine =9.9818 log sine =9.9818 

/. c= 73o 32.5', log sin ,1 = 9^916 
or 106° 27.5'. 

Am. l.A= 78° 46.7', a = 70° 10', c = 106° 27 5' 

2. A = 101° 13.3', a = 109° 50', c = 73° 32.5'. 



17. t2inA= cotB -. sin 6 = sin 5 sine. 

cosc log sin £=9.9681 

log cot B= 9.5998 logsinc = 9 . 8681 

log cosc =9.8291 

— . log sin b =9.8362 

log tan .4=9.7707 ... h = 43 o 17 9 , 

.*. ^4 = 30° 32.1'. 

(7Aec&. 
tan a = cos B tan c. tan a 

log cos B = 9.5679 tan A = ^y 

log tan e = 0.0389 log tan a = 9mm 

log tan a = 9.6068 lo S sin & = 9.8362 

"•"■ a = 2 2°l.l'. log tan .4 =9.7706 



1Q * ~ + 

A o. tan a = tan .4 sin b. 

log tan .4=9.5152 
log sin b = 1 



log tan a =9.3921 
180° -a =13° 51.3'. 
/. a = 166° 8.7'. 



cosi?= sin ^4 cos 6. 
log sin A = 9.4931 
log cos 6 =9.8181 



log cos 5 = 9.3112 
180° - J9= 78° 11.1'. 

.'. 5= 101° 48.9'. log cos B= 03112 



+ 

tanc = 


tan b 
cos A 


log tan b = 
log cos A = 


= 0.0588 
= 9.9779 


log tanc = 

.'. c = 


= 0.0809 
= 50° 18.4'. 


Check. 




cosB = 

log tan a = 
log tanc = 


tana 
tan c 
9.3921 
0.0809 - 



68 



KEY TO ESSENTIALS OF TRIGONOMETRY. 



19. 



tan A = 



tan a 



20. 



sin b 
+ 
log tan a = 0.3634 
log sin b =9.9707 

log tan A = 0.3927 

180°-4 = 67°57.5'. 
/. 4 = 112° 2.5'. 

+ - - 

cosc = cos a cos b. 

log cos a = 9.5993 
log cos & =9.5500 

log cose =9.1493 
.-, c = 81° 53.6'. 

- - + 

cos A = cos a sin B. 

log cos a = 9.8652 
log sin B = 9.9846 

log cos A =9.8498 
180° - A = 44° 57.5'. 
.-. 4 =135° 2.5'.. 



tan b = sin a tan i?. 
log sin a =9.8325 
log tan B =0.5674 

log tan b =0.3999 
.-. b = 68° 17.3'. 



, -r, tan 6 

tan2?= 

sin a 

+ 

log tan b =0.4207 

loffsina =9.9626 



log tan B =0.4581 
180° -B= 70° 48'. 
.-. B =109° 12'. 

Check. 
cos c tan A tan 5=1. 

log cose =9.1493 
log tan A =0.3927 
log tan B = 0.4581 



log 1 = 0.0001 



tanc : 



tan a 



cos B 

+ 
log tan a =9.9674 
log cos B = 9.4172 

log tanc =0.5502 

180° -c =74° 16'. 
.-. c=105°44'. 

Check. 

A tan b 

cos A = 

tanc 

log tan b =0.3999 

log tanc =0.5502 

log cos A = 9.8497 



21. 



cos A 

cos a = • 

sin B 

+ 

log cos A =9.9129 

log sin B = 9.9916 

log cos a =9.9213 
180° -a = 33° 27.5'. 
.-. a =146° 32.5'= 



cos b = 



cos B 



sin A 

+ 
log cos B = 9.2896 
log sin A = 9.7597 

log cos b =9.5299 
180° -b = 70° 11.9'. 
.-. 6= 109° 48.1'. 



CHAPTER XL— PAGES 112, 113. 69 



22. 



+ - - 

cos c = cot A cot B. 


Check. 


log cot A = 0.1532 


cose = cos a cos b. 


log cot B =9.2080 


log cos a =9.9213 


log cose =9.4512 
.-. c = 73°35'. 


log cos b =9.5299 


log cose =9.4512 


A sin a 


7 cos c 

cos = 



log 


cos b = 


: 9.4371 




.'. b = 


: 74° 7.3'. 




Check. 






sin^L = 


cos B 
cos b 


log 


cos B = 


9.4101 


log 


cosb — 


9.4371 



sin c cos a 

log sin a =9.9710 log cose =8.9855 

log sine =9.9980 log cos a =9.5484 

log sin A = 9.9730 
.-. A =70°. 

-d tan a 

cos B = *■• 

tan c 

log tan a = 0.4226 
log tan c = 1.0125 

log cos B =9.4101 

.-. B = 75° 6.2'. log sin A = 9.9730 



Art. 154. — Page 113. 

2. In the polar triangle, a' = 41° and B' = 37°. 

cos A' = cos a f sin B' . tan b f = sin a' tan B f . 

log cos a' =9.8778 log sin a' =9.8169 

log sin B f = 9.7795 log tan B' = 9.8771 

log cos A' =9.6573 
.-. A = 62° 58.8'. 

tane<: - tana ' 



cosB' 



log 


tan 6' 


= 9.6940 




.-. b' 


= 26° 18.1'. 




Check. 




cos A' 


_ tan b' 



log tan a' =9.9392 tanc' 

log cos B' = 9.9023 log tan b' =9.6940 

log tan c' = O0369 lo S tan c ' = °- 0869 

.-. c' = 47° 26'. log cos A 1 = 9.6571 

.*. in the quadrantal triangle, 

a =117° 1.2', £=153° 41.9', and 0=132° 34'. 



70 KEY TO ESSENTIALS OF TRIGONOMETRY. 

3. In the polar triangle, a ( = 134° 30' and b> = 40° 40'. 



tan A 1 = 



tana' cos c' = cos a' cos b'. 

sin 6' log cos a' =9.8457 

+ log cos 6' =9.8800 



log tan a' =0.0076 logcogc , =Mm 

log sin V = 9.8140 >% 1SQO _ c , = 5?0 ^ ^ 

log tan .4' =0.1936 

,-. 180° -.4' = 57° 22.1'. 

Chech. 

t an y cos c' tan A! tan _B' = 1. 

tan B 1 = 

sin a' log cose' =9.7257 

log tan b f =9.9341 log tan A' = 0.1936 

log sin a' =9.8532 log tan JB' = 0.0809 

log tan B> = 0.0809 log 1 = 0.0002 

.-. B ! = 50° 18.4'. 

in the quadrantal triangle, 

a =57° 22.1', 6 = 129° 41.6', and (7= 57° 52.5'. 



4. In the polar triangle, A' = 149° 40' and c'= 137° 20'. 

sin a' = sin A' sin c'. + — — 

i a i nirnoo tan 6' = cos JL' tan c'. 

log sin J.' = 9./ 033 

log sine' =9.8311 log cos -4' = 9.9361 



log sin a' =9.5344 



log tan c' =9.9646 



180° -a' = 20° 0.9'. log tan 6' =9.9007 

.-. b' = 38° 30.4'. 



cot^L' 



Check. 



tani?' = , -r,, tan 6' 

cos c ' tan B 1 = -• 

sin a' 

log cot JL' = 0.2327 log tan 6' =9.9007 

log cose' =9.8665 log sin a' = 9.5344 

log tan B' = 0.3662 log tan B' = 0, 

.-. ^' = 66° 42.9'. 

/. in the quadrantal triangle, 

^ = 20° 0.9', b =113° 17.1', and B =141° 29.6'. 



CHAPTER XL — PAGE 113. 71 

5. In the polar triangle, b' = 109° 48' and c' = 73° 35'. 

_ + - 

cosa' = ™l£!. - •' tan ^ 

cos 6' 

log cose' =9.4512 
log cos V =9.5299 







tan c ' 






+ 


log tan 


V 


= 0.4437 


log tan 


c' 
A' 


== 0.5307 


log cos 


= 9.9130 


180°- 


A 


= 35° 4.4'. 




Check. 


sin 


P> ! 


cos A' 



log cos a' =9.9213 
/. 180° -a' = 33° 27.5'. 

-o, sin b' 

sm B' = -• 

sine 7 

log sin b' =9.9735 cos a' 

log sine' =9.9819 log cos -4' = 9.9130 

i • -o/ a nniij log cos a =9.9213 
log sin B' = 9.9916 & . 

... 180° - B' = 78° 46.7'. log sin B' = 9.9917 

.*. in the quadrantal triangle, 

a = 35° 4.4', ^.= 33° 27.5', and b =78° 46.7'. 

6. In the polar triangle, a' = 74° V and A' = 75° 6'. 

. , , tan a f 

sm o' = • 

tan J.' 

log tan a' = 0.5459 

log tan A f = 0.5750 

log sin b' =9.9709 
.-. b' = 69° 16', 
or 110° 44'. 

, sin a' 

sin c' = 

sin^l' 

log sin a' =9.9831 
log sin A f = 9.9851 





sin.B' 


cos .4' 
cos a' 


log 


cos A' 


= 9.4102 


log 


cos a' 


= 9.4372 


log 


sin J B / 


= 9.9730 




.'. B' 


= 70°, 




or 110°. 




Check. 




sinB r 


sin b' 
sin c' 


log 


sin b' 


= 9.9709 


log 


sine' 


= 9.9980 



log sine' =9.9980 

.-. c' = 84° 30', log sin B , _ 9 9729 

or 95° 30'. 

.-. in the quadrantal triangle, 

1. B= 69° 16', 6= 70°, and C= 84° 30'. 

2. £=110° 44', 6=110°, and C=95°30'. 



72 KEY TO ESSENTIALS OF TRIGONOMETRY. 



CHAPTER XII. 

Art. 167. — Page 125. 

2. Here J (J. - B) = 18° 30', J (4 + B) = 59° 30', J c = 54°. 

tanK«-^)= Sin ' ( ^""^ ) tanic. tan |(a + 5) = c _^iMz^l tan i, 

log sin J (.4 - 5) = 9.5015 log cos i(A - B) = 9.9770 

log esc i(A + B)= 0.0647 log sec i (A + B) = 0.2945 

log tan i c =0.1387 log tan J c =0.1387 

log tan J (a - 6) = 9.7049 log tan J (a + 5) = 0.4102 

... J( a -&) = 26°52.8'. .\ J (a + 6) = 68° 45'. 

.-. a = J (a + b) + J (a - &) = 95° 37.8', 
and • b = J (a + 6) -J (a -6) = 41° 52.2'. 



^i^sjnK^) 

sin J (a -6) 2V ' 



log sin § (a + 6) =9.9694 
log esc J (a -6) =0.3447 
log tan h(A-B) =9.5245 

log cot J C = 9.8386 

i(7= 55° 24.4' 
.-. O=110° 48.8'. 

3. Here i(B- <?) = 42° 30', J(5+ C) = 92°30', Ja = 35°10'. 

tan }(6 - c)= sin ^^-°) tan Ja. tanj(6 + = C0S i( J - ^ tan ' o 
2V y sinK-B+C') cosK^+O) " 

log sin J (B - G) = 9.8297 - 

log esc J (JB + (7) = 0.0004 lo S cos I (-B - <?) = 9 - 8676 

log tan J a =9.8479 log sec J (^+ C) = 1.3603 

log tan £ a =9.8479 

log tan j (& -c) =9.6780 " 

... J(/;-c).= 25°28.5'. log tan J (6 + c) =1.0758 

180° -i (b + c) = 85° 11.9' 
.-. J(6 + c) = 94°48.1/. 
... 6 = J (6 + C ) + | (6 - c) = 120° 16.6', 
and c = } (6 + c) - J (»- c) = 69° 19.6'. 



CHAPTER XII. — PAGE 125. 73 

cot \A = s ! n Kf + c ) tan \(B-C\ 
sin \ (b — c) 

logsin£( 6 + =9.9985 

log esc J (6 - c) = 0.3664 

logtanK^-ff) = 9.9621 

log cot J J. =0.3270 

\A = 25° 13.1' 

.-. ^4 =50° 26.2'. 

4. Here J(0-^) = 45°20', }(#+ ^) = 77°, J6 = 20°20'. 

tanKc-fl)= 8in ^ C "^ tan ^6, tani(c + a)= C0S ^ C ""^ tanU. 

log sin J ((7- -4) = 9.8520 log cos ±(C - A) = 9.8469 

log esc i ( C + 4) = 0.01 13 log sec J ( C + A) = 0.6479 

logtanjfc = 9.5689 logtanJ6 = 9.5689 

logtanJ( c -«) =9.4322 log tan | ( c + «) =0.0637 
... J ( c - a) = 15° 8.2'. .-. i(c + d) = 49° 11.2'. 

... a = i(c + a)-i(c-a) = S4?S', 
and c = J (c + a) + \ (c - a) = 64° 19.4'. 



sin J (c — a) 



log sin J ( c + a ) = 9.8790 
log esc i (c — a) = 0.5831 
log tan i (C-A) = 0.0051 
log cot J B =0.4672 

J 5 =18° 49.8' 
.-. B = 37° 39.6'. 

5. Here J (5 - 4) = 18° 47', J (5 + ^) = 126° 59', J c = 63° 16'. 

tan 1 (h a\- sin ^ ^ "" =J tan I c + 4- 

^ a) -sin K B + ^) tan2C - .fnKM-a^^t^-^tA 
log sin J (J5 - ^) = 9 - 5 °79 C0S ^ (^ + ^) 

log esc J (2* + A) = 0.0976 cqs }(i? __ = Q 9?62 

log tan i c = 02979 gec = 

logtanK6-a) =9.9034 ^^ ^^ 

,.»<*- «) = 38040.8'. lo g ta 4 (6 + a) = 04048 

180°-} (6 + a) =72° 15.2' 
.-. i( & + a ) = 107 ° 44 - 8 '- 
.'.«» J.(5 + a) - J (ft -a) = 69° 4', 
and 6 = j (6 + a) + } (6 - a) = 146° 25.6'. 



c. 



74 KEY TO ESSENTIALS OF TRIGONOMETRY. 



cot | C = sin *(>+<») tan i(S-A). 
sin J (6 — a) 



log sin J (6 + a) = 9.9788 
log esc i(b — a) = 0.2042 
log tan J (B - 4) = 9.5316 

log cot J C =9.7146 

JC = 62° 36.1' 
.-. (7=125° 12.2'. 

Art. 168. — Page 126. 

2. Here J (a - 6) = 12° 30', § (a + 6) = 59° 30', }C = 16° 30'. 

tanK^-i?)= SinKa ~~ 6) coUa tanK^+^) = COgKa "" 6) cot*a 

sin ^ (« + b) ' cos £ (a + 6) 

log sin J (a - 6) = 9.3353 log cos J (a - 6) = 9.9896 

log esc i(a + b) = 0.0647 log sec J (a + 6) = 0.2945 

log cot JO = 0.5284 log cot J C =0.5284 

log tan I (4 — B) = 9.9284 log tan J (J. + £) = 0.8125 

... £ (4 - .B) = 40° 18'. .-. \(A + B) = 81° 14.8'. 

.-. J. = J (J. + B) + H^ ~ B) = 121° 32.8', 
and 5 = J 04 + JS) - i(A - 5) = 40° 56.8'. 

tan | c = sin 2 (^ + -B) tan l (a - 6) . 

2 sin J (4. - J5) 2 v ' 

log sin J O^ + ^) =9.9949 
log esc i(A-B) =0.1892 
log tan J (a -6) =9.3458 

log tan J c =9.5299 

\ c = 18° 42.9' 
.-. c = 37° 25.8'. 

3. Here ±(a-c) = 19°, ±( a + c ) = 79 °> i B = 55°. 

t&nHA-C)= sini ( a ~^ cotiB. tanK^ + 0) = cos ^ a ~ c:) cotJP. 
sinj(a + c) cos J(a + c) 

logsinJ(«-c) =9.5126 log cos J (a - c) =9.9757 

log esc i (a + c) = 0.0081 log sec J ( a + c ) = 0.7194 

log cot J B =9.8452 log cot i B =9.8452 

log tan i (A— O) = 9.3659 log tan J (4 + <!) = 0.5403 

.-. | (A - C) = 13° 4.4'. .-. } (A + C) = 73° 55.3'. 

/. 4 = £(^ + C) + K^-^) = 86°59.7', 

and C = } (-4 + 0) - J O^ - C) = 60° 50.9 . 



CHAPTER XII. — PAGE 126. 75 

, , j sin o (A + C) . i , n 

tan £ b = 2_v — i 1 tan £ (a — cY 

sin J (A — C) 
log sin J (A+ C) = 9.9827 
log esc i (A — C) =0.6455 
log tan i (a - c) = 9.5370 
log tan J 6 =0.1652 

}&= 55° 38.5' 
.-. b =111° 17'. 

4. Here }(fr — c) = 24 ° 50 ', K 6 + c ) = 95 ° 30 '> M = 25°. 

U*HB-C) = ^^ co H A. - co g K6-c) + 

sm 2 {o-\-cj tani(2? + C) = £- (-cot J A 

logsin£(&-0 =9.6232 coaJ(6 + c) 

log esc I (b + c) = 0.0020 "~ 

log cot* -A =0.3313 logcosK6-c) =9.9579 

log tan *(B-C) = 09566 log sec J (* + =1-0184 

• ±(B—C)=42°SV log cot £ J. = 0.3313 

' 2 ^ J ' log tan \ (B + C) = 1.3076 

180°- J (5+ C) = 87°10.8' 
.-. J (5+ C) = 92°49.2'. 

/. £=|(£ + #) +1(^—0) = 134° 57.3', 
and C=J(B+C) -1(^-0)= 50° 41.1'. 

tan2a -sinK^-^) 2( } * 

log sin J (5 + C) = 9.9995 
log csc J (-#-<?) =0.1734 
log tan i(b — c) = 9.6664 
log tan J a = 9.8393 

Ja = 34°34.4' 
.-. a = 69° 8.8'. 

5. Here § (6 - a) = 14° 20', \ (b + a) = 139° 30', } C= 70° 10'. 
tanK£-^)= . -;. , ; coU(7. - CO oiTa__^ + 

log sin J (6 -a) =9,3937 cos _ (6 -fa) 

log esc Ub + a) = 0.1875 ~~ 

log cot J C =9.5571 lag ca. *(*-«) =9.9863 

logtanK5-^l) = 9l383 Jog «*£(* + «) =0.1190 

.-. i ( 5 - ^) = 7° 49.8'. log cot - ° = 95571 

2 v log tan ; (B + A) = 9.6624 

180° ~-i(B+A) = 24°41.2' 

/. J(^ x ^) =155° 18.8'. 
.-. .4= KB 4- ,4) _i(^_^) = i47 o 20', 
and B = i(B + A ) + K-B - -4) = 163° 8.6'. 



76 KEY TO ESSENTIALS OF TRIGONOMETRY. 



tani c = ™l^±41tanj(&-a). 



\ogsin%(B+A) =9.6208 
log esc i (B-A) =0.8657 
log tan J (6 - a) = 9.4074 

log tan J c =9.8939 

}c=38°4.2' 

.-. c = 76° 8.4'. 



Art. 169. — Page 127. 

2. Here s = 65° 30', s ~ a = 27° 30', s - 6 = 14° 30', s - c = 23° 30'. 

log sin (s- 6) =9.3986 log sin (s - c) =9.6007 

log sin (s-c) = 9.6007 log sin (s - a) = 9.6644 

log esc s =0.0410 log esc s =0.0410 

log esc (s — a) =0.3356 log esc (s - b) =0.6014 

2 )9.3759 2 )9.9075 

log tan %A =9.6879 log tan J B =9.9537 

J^ = 25°59.1' J^ = 41°57.2' 

.-. A = 51° 58.2'. .-. B = 83° 54.4'. 

log sin (5 -a) =9.6644 

log sin (s- b) =9.3986 

log esc s =0.0410 

log esc (s-c) = 0.3993 

2 )9.5033 
log tan J C =9.7516 
i 0=29° 26.6' 
.-. C= 58° 53.2'. 

3. Here s = 105°, s - a = 4°, s - b = 56°, 5 - c = 45°. 

log sin (s - 6) = 9.9186 log sin (s-c) = 9.8495 

log sin (s-c) = 9.8495 log sin (s - a) = 8.8436 

log esc s =0.0151 log esc s =0.0151 

log esc (s-a) = 1.1564 log esc (s - 6) = 0.0814 

2 )0.9396 2 )8.7896 

log tan J ^4 =0.4698 log tan J J5 =9.3948 
iA= 71° 16.4' } 5 =13° 56.3' 

.-. A = 142° 32.8'. .-. B = 27° 52.6'. 



CHAPTER XII. — PAGE 127. 77 

log sin (s - a) = 8.8436 
log sin (s- 6) =9.9186 
log esc s =0.0151 

log esc (s — c) =0.1505 

2 )8.9278 
log tan J C =9.4639 

£(7=16° 13.6' 
/. 0=32° 27.2'. 



4. Here s = 96°, s-a = 35°, s - 6 = 57°, s - c = 4°. 

log sin (s- b) =9.9236 log sin (s-c) =8.8436 

log sin (s-c) =8.8436 log sin (s - a) =9.7586 

log esc s =0.0024 log esc s =0.0024 

log esc (s — a) =0.2414 log esc (s — b) =0.0764 

2 )9.0110 2 )8.6810 

log tan iM =9.5055 logtan|J5 =9.3405 

1.4 = 17° 45.6' \ B=12° 21.3' 

/. A = 35° 31.2'. .\ B = 24° 42.6'. 



log sin (s- a) =9.7586 

log sin (s- b) =9.9236 

log esc s = 0.0024 

log esc (s — c) = 1.1564 

2 )0.8410 
log tan }C =0.4205 
|C=69°12.4' 
.-. C= 138° 24.8'. 



5. Here s=107°10', s-a = 44°50', s-6 = 53°, s-c = 9°20'. 

log sin (s- b) =9.9023 log sin (s-c) =9.2100 

log sin (s-c) =9.2100 log sin (s - a) =9.8482 

log esc s =0.0198 log esc s =0.0198 

log esc (s — a) =0.1518 log esc (s — 6) =0.0977 

2 )9.2839 2 )9.1757 

log tan \ A =9.6419 log tan \ B =9.5878 
\A = 23° 40.6' J 5 = 21° 9.7' 

.-. .4 = 47° 21.2'. „\ ^ = 42° 19.4'. 



78 KEY TO ESSENTIALS OF TRIGONOMETRY. 

log sin (s- a) =9.8482 

log sin (s — b) =9.9023 

♦ log esc s =0.0198 

log esc (s-c) =0.7900 

2 )0.5603 
log tan J (7 =0.2801 
JO=62°19' 
.-. C =124° 38'. 



Art. 170. — Page 128. 

2. Here #=109°, #-.4 = 34 , S-B = 27°, £-0=48°. 

log cos S =9.5126 log cos S =9.5126 

log cos (#- A) = 9.9186 log cos ($- B) = 9.9499 

log sec (S- B) = 0.0501 log sec (#- C) = 0.1745 

log sec (£ - C) = 0.1745 log sec (# - A) = 0.0814 

2 )9.6558 2 )9.7184 

log tan J a =9.8279 log tan J b =9.8592 

Ja = 33°55.9' i& = 35°52.2' 

.-. a = 67° 51.8'. /. b =71° 44.4'. 

log cos S = 9.5126 

logcos(#- C) = 9.8255 
log sec (S- A) =0.0814 
logsec(^--B) =0.0501 

2)9.4696 
log tan J c =9.7348 

i c = 28° 30' 
.\ c = 57°. 



3. Here #=165°, £-.4 = 45°, £-£=35°, £-0=85°. 

log cos # =9.9849 log cos S =9.9849 

log cos (S-A) = 9.8495 log cos (£- 5) = 9.9134 

log sec (£-£) = 0.0866 log sec (tf - O) = 1.0597 

log sec Is- O) = 1.0597 log sec (S—A) = 0.1505 

2 )0.9807 2 )1.1085 

log tan J a =0.4903 log tan J 6 =0.5542 

Ja= 72° 4.9' J6= 74° 24,3' 

.\ a = 144° 9.8'. .\ b = 148° 48.6'. 



CHAPTER XII. — PAGE 128. 79 

log cos S = 9.9849 

logcos(#- C) = 8.9403 
log see (S— A) =0.1505 
log sec (S- B) =0.0866 

2)9.1623 
log tan i c =9.5811 

£c = 20°51.8' 
,\ c = 41° 43.6'. 



4. Here #=124° 40', S - A = 33° 30', S-B = 39°, tf- C= 52° 10'. 

log cos tf =9.7550 log cos S =9.7550 

log cos (S-A) = 9.9211 log cos (S - B) = 9.8905 

log sec (£ - jB) = 0.1095 log sec (S - C) = 0.2123 

logsec(#-C) = 0.2123 log sec (S— A) = 0.0789 

2 )9.9979 2 )9.9367 

log tan J a =9.9989 log tan J 6 =9.9683 

| a = 44° 55.6' J 6 = 42° 54.6' 

.-. a = 89° 51.2'. .-. b = 85° 49.2'. 

log cos S =9.7550 

* logcos(£-C) = 9.7877 
log sec (S- A) =0.0789 
log sec (S-B) =0.1095 

2 )9.7311 
log tan \ c = 9.8655 

Jc = 36°15.9' 
.-. c = 72° 31.8'. 



5. Here 

£=102° 40', #-.4 = -35 36', S-B=7l°29', #-C=66°47'. 

log cos S =9.3410 log cos S =9.3410 

log cos (S-A) = 9.9102 log cos (S - B) = 9.5019 

log sec (S-B) = 0.4981 log sec (S- C) = 0.4043 

log sec ls-C) = 0.4043 log sec (S-A) = 0.0898 

2 )0.1536 2 )9.3370 

log tan J a =0.0768 log tan J b =9.6685 

J« = 50°2.3' J6 = 24° 59.4' 

.-. a = 100° 4.6'. .-. b = 49° 58.8'. 



2)9.5246 








= 9.7623 








}c = 30°3' 








. c = 60°6'. 








. — Page 131. 








KM- 


c) 


= 82°, 




i(P- 


C) : 


= 17° 40'. 




i(B+C) 


= 80° 35', 




i(B- 


-G) 


= 15° 5'. 



KEY TO ESSENTIALS OF TRIGONOMETRY. 

log cos S = 9.3410 

log cos (£- (7) = 9.5957 
logsec(#-^) =0.0898 
log sec (S-B) = 0.4981 



log tan J c 



A . ~ sin c sin 2? 
4. sin O = — 

sin 6 

log sin c = 9.9549 
log esc b = 0.0062 
log sin B = 9.9979 
log sin (7= 9.9590 
.\ 0=65° 30'. 

1 sin^(6-c) JV y sin £(£—(7) 

log sin J (6 + c) = 9.9958 log sin \ (B + G) = 9.9941 

log esc i(b — c) = 0.5179 log esc J (B - (7) = 0.5847 

log tan HB - C) = 9.4306 logtanJ(6-c) = 9.5031 

logcoti^d =9.9443 log tan J a =0.0819 

%A = 48° 40' £a = 60°22.3' 

.-. .4 = 97° 20'. .-. a = 100° 44.6'. 

5 sin B = sin b sin - « J (6 + a) =79° 10', 

sin a ' K&-<0 =39° 10'. 

log sin 6 = 9.9446 J (5 + ^) = 36° 10', or 83° 30'. 

log esc a = 0.1919 h(B-£)= 6° 30', or 53° 50'. 

log sin A = 9.6946 2K J 
log sin 5 =9.8311 

.-. jB=42°40', or 137° 20'. 

coH(7= sinK6-fa) ta ^ ^ t an}c = s l D ^g + ^ tanK^a). 

sin J (6 — a) sinJ(Z? — ^1) 

Using the first value of B, we have : 

log sin i(b + a) = 9.9922 log sin | (5 + A) = 9.7710 

log esc | C 6 — «) = - 19 ^6 log esc J (5 — ^1) = 0.9461 

log tan \(B-A) = 9.0567 log tan J (^- a) = 9.9110 

log cot i C =9.2485 log tan J c =0.6281 

\ C= 79° 57' Jc=76°45.1' 

.-. C = 159° 54'. /. c = 153° 30.2'. 



CHAPTER XII. — PAGE 131. 81 

Using the second value of B, we have : 

log sin J (&+<i) =9.9922 log sin -J (B + A) =9.9972 

logcsc£(6-a) =0.1996 log esc l(B- A) =0.0930 

log tan % (B — A) = 0.1361 log tan] (6 -a) = 9.9110 

log cot JO =0.3279 log tan J c =0.0012 

JC=25°10.3' Jc = 45°4.8' 

/. C= 50° 20.6'. .-. c = 90°9.6'. 

6. sin^= sinasinC . 

sine 

log sin a =9.9561 
log esc c =0.2562 
log sin C = 9.7973 
log sin A = 0.0096 
Since log sin A is positive, the triangle is impossible. 

7. sinC= sincsin ^ 



sin a cos.B=^£. 

log sine =9.9958 tan c 



log esc a = 0.0252 



+ 



i • a n htaa log tan a = 0.4549 
log sin A = 9.9/90 fo 

i • n a nnnn log tan C = 0.8522 

log sin (7 = 0.0000 & 

. p_o()o log cos 5= 9.6027 

~" ' 180°- J3= 66° 23.1' 

.-. B= 113° 36.9'. 



COS 6: 



+ 
COS c 



log cos c = 9.1436 
log cos o= 9.5199 
log cos 6 = 9.6237 
180°-6 = 65°8.1' 
.-. 6= 114° 51.9'. 

8 sinJB= sin6sinC *<* + *) =74° 40', 

sine i(b-c) =33° 50'. 

log sin b = 9.9770 + = 540 Qr ^ Q ^ t 

log esc e =0.1845 _ , 

log sin (7= 9.8066 2 \ ' 

log sin B= 9.9681 

.-. 5 = 68° 18', or 111° 42'. 

cotM=$4^tanK£-C). tan Ja = $lM|±^ tan K^-c). 
sinj(6 — c) sin£(#— Gj 



82 KEY TO ESSENTIALS OF TRIGONOMETRY. 

Using the first value of B, we have : 

log sin J (b + c) = 9.9843 log sin | (B + C) = 9.9084 

log esc i(b- c) = 0.2543 log esc J (B - C) = 0.6093 

log tan %(B — C) = 9.4042 log tan J (6 — c) = 9.8263 

log cot J A =9.6428 log tan J a =0.3440 

iA = 66° 16.9' Ja = 65 c 37.9' 

.-. J. = 132° 33.8'. /. a = 131 c 15.8'. 

Using the second value of B, we have : 
log sin ^(b + c) = 9.9843 log sin J (B + C) = 9.9865 

log esc } (ft — c) = 0.2543 log esc J (JB - C) = 0.2315 

log tan J (JB — C) = 9.8602 log tan J (6 - c) = 9.8263 

log cot J J. =0.0988 log tan J a =0.0443 

1^4 = 38° 32.3' Ja = 47°55' 

.-. 4=77° 4.6'. /.a = 95° 50'. 

n . . sin a sin B 
9. sin ^4 = 

sin 6 

log sin a = 9.4821 
log esc b = 0.5686 
log sin B =9.9134 

log sin .4=9.9641 

.-. 4 = 67° 1.7', or 112° 58.3'. 
Since both values of A are < B, while a is given > b, the triangle is 

impossible. 

10 cinC= sincsin ^ K c + «) =96° 35', 

sin a K c - a ) =41° 35'. 

log sin c = 9.8241 i ( q + A) = 94° 33.95', 

log esc a = 0.0866 i ( q - A) = 52° 3.95'. 
log sin ^ = 9.8297 

log sin (7=9.7404 
.-. C= 146° 37.9'. 

co sinj_(c + a) tan tan}6 = sin ^ C+ ^ tanK^a). 

log sin K c + «) = 9.9972 log sin J ( O + -4) = 9.9987 

log csc J (c- «) =0.1780 log esc J (C- -4) =0.1031 

log tan } ( C - A) = 0.1082 log tan i (c - a) = 9.9481 

log cot \ B =0.2834 log tan J b =0.0499 

J£=27°30.3' J6 = 48°17.2' 

/. B = 55° 0.6'. /. b = 96° 34.4'. 



CHAPTER XII. — PAGE 133. 83 



Art. 172. — Page 133. 

sin b = s[nBsinr i (B+C) = 98°, 

sin C i( B ~C)=lS°. 
log sin 2? = 9.9537 ^99° 25', 

log esc C= 0.0066 



log sine = 9.9976 
log sin 6 =9.9579 
.-. 6 =114° 50'. 



i(6_c) =15° 25'. 



cot^= sin ^ 5+c )tani(^-C). tanJa=^i(^L^l tan i (6 _ c) . 

sinj(^-c) - V J sini(B-C) 2V J 

log sin J(6 + c) = 9.9911 log sin § (# + C) = 9.9958 

log esc I (b-c) = 0.5751 log esc J (J5 - C) = 0.5100 

log tan K^- C)= 9.5118 log tan £ (6 -c) = 9.4405 

log cot § A =0.0813 log tan J a =9.9163 

J^ = 39°10' |a = 41°28' 

.-. ^L=79°20'. /.a = 82° 56'. 

3 gin a = sin A sin 6 \ (B + A) = 136°, 

logsin^l = 9.8711 |(6 + fl) ==9 7o 12 , )0rll9 o 48 , < 

log esc 5=0.1919 Kh _ a) =29 48',or7<>12'. 

logsinZ) = 9.9023 2 ^ J 
log sin a = 9.9653 

.-. a = 67° 21', or 112 c 36'. 

smi(6 + a) teie= ii»KJ+i) tot( i, fl)i 

2 an £(6 -a) sinJ(-B--4) 
Using the first value of a, we have : 

log sin J (6 + a) = 9.9966 log sin \(JB+A) = 9.8418 

log esc J (6 - a) = 0.3036 log esc $ (B — A) = 1.1561 

log tan j(B — A) = 8.8116 log tan |(6 — «) = 9.7579 

log cot JO =9.1118 log tan J c =0.7561 

JC= 82° 3.2' Jc = 80°3.2 

/. C= 161° 6.1'. .\ c= 160° 6.1'. 

Using the second value of a, we have : 

log sin l(h + a) = 9.9381 log sin \ (B + A) = 9.8118 

log esc i(b- a) = 0.9019 log esc \ (B — A) = 1.1561 

logtanH-B-^) = 8.8146 log tan j (6 — a) = 9.1015 

log cot J C =9.6819 log tan \ c =0.0997 

£C=64°10.3' Jc = 51°31.2 ; 

,\ C = 128° 20.6'. .-. c = 103° 2.1'. 



84 KEY TO ESSENTIALS OF TRIGONOMETRY. 

A . sin C sin a 

4. sin c — • 

sin A 

log sin C = 9.9904 

log esc A = 0.0541 

log sin a = 9.9555 

log sine =0.0000 

.-. c = 90°. 

The triangle is a quadrantal triangle, and the sides a' and c ! of its 

polar triangle are 118° and 78°. 

_ + _ "-* 

j, cose' Ty, tana' 

cos 6' = -• cos B' = 

cos a' tanc' 

- + 

log cos c' = 9.3179 log tan a' = 0.2743 

log cos a' = 9.6716 log tanc' = 0.6725 

log cos V = 9.6463 log cos B 1 = 9.6018 

.-. 180° - b> = 63° 42.7'. /. 180° - B' = 66° 26.2'. 

Therefore in the given triangle, B= 63° 42.7', and 6 = 66° 26.2'. 

K . 7 sin B sin a 

o. sin o = • 

sin ^4 

log sin B = 9.9624 
log esc .4 = 0.1418 
log sin a = 9.9948 
log sin b =0.0990 
Since log sin b is positive, the triangle is impossible. 

6 gin b = sin B sine J (C + B) = 84 c 30', 

sinC J(C-J5) = 62°10'. 

log sin B = 9.5798 J ( c + 6 ) =82° 51.05', 

log esc C = 0.2600 j ( c - 6 ) = 55 ° 28.95'. 

log sine = 9.8227 
log sin 6 =9.6625 
.-. 6= 27° 22.1'. 

coU4 = sin ^ c + 6 > tanKO-i?). tan»a= sin ^ C+ J) tanKc-6). 
sin J(c — o) sinj((7— -B) 

log sin } (c + 6) = 9.9966 log sin J ( G + B) = 9.9980 

log esc } (c - 6) = 0.0841 log csci(C-B) = 0.0534 

log tan | ((?- B) = 0.2774 log tan J (c- 6) = 0.1626 

log cot f A =0.3581 log tan J a =0.2140 

J A = 23° 40.6' J a = 58° 34.6' 

/, A = 47° 21.2'. .-. a = 117° 9.2'. 



CHAPTER XII. — PAGES 133, 134. 85 

7 sina = sin ^ sinc . HC+A) = lOl°, 

sinO j(C-^l) = 39 o 20 / . 

log sin .4 = 9.9446 £(c + a) =90° 41.55', 
log esc C= 0.1950 or 143° 38.45'. 

log sine = 9.6946 J(c-«0 -53° 38.45', 
log sin a =9.8342 or 6° 41.55'. 

.-. a = 43° 3.1', or 136° 56.9'. 

2 sinj(c-a) ^ y sinJ(C-^l) 

Using the first value of a, we have : 

log sin i(c + a) = 9.9971 log sin J ( C + A) = 9.9919 

logcsc£( c -«) =0.0940 log esc J (C- 4) = 0.1980 

logtan}(C-4) = 9.9135 log tan J (c - a) =0.1330 

log cot i B =0.0046 log tan J b =0.3229 

J_E=44°41.9' f& = 64° 34.2' 

.-. J5=89°23.8'. .-.6=129° 8.4'. 

Using the second value of a, we have : 

log sin K c + «) = 9 -?730 log sin J ( C + A) = 9.9919 

log esc J (c — a) = 0.9335 log esc J (C— ^) = 0.1980 

log tan J (C-^) = 9.9135 log tan £ (c - a) =9.0695 

log cot \ B =0.6200 log tan J 6 =9.2594 

J 5=13° 29.3' £6 = 10° 17.9' 

.-. B = 26° 58.6'. .-. b = 20° 35.8'. 

8 sin C sin b 

sin c = - 



sini? 
log sin (7 = 9.9950 
log esc B = 0.0194 
log sin 6 =9.9252 

log sine =9.9396 

.-. c = 60° 28.6', or 119° 31.4'. 
Since both values of c are < b, while C is given > B, the triangle is 
impossible. 

Art. 173. — Pages 134 and 136. 

1. Let B and G denote the positions of Boston and Greenwich, 
respectively, and P the north-pole. 

Denote the arcs PG, PB, and BG by b, g, and p. 



86 KEY TO ESSENTIALS OF TRIGONOMETRY. 

Then in the triangle PBG, 

ZP=71°4', b = 90° -51° 29' = 38° 31', #= 90° -42° 21' = 47° 39'. 
By Art. 162, we have : 

sin i (g + v 

tanKg + J) = C ° g ii g ~S cot^P. 

cos i(g + o) 
From the data, 

i(?-6) = 4°34', JGr+6) = 43°5' f JP=35°32'. 
log sin i(g-b) = 8.9009 log cos J (^ - b) = 9.9987 

log esc } O + 6) = 0.1656 log sec J (# + 6) = 0.1365 

log cot J P =0.1462 log cot £P =0.1462 

log tan J ( 6? - P) = 9.2127 log tan } (© + B) = 0.2814 

... £(0-5) = 9° 16.1'. .-. }(^ + ^) = 62°23.2'. 

.-. P = K^+P)-K^--B)-53°7.1', 
and G= i (G + B) + £(#- #) = 71° 39.3'. 

tan |ji = !HLii^_±^) tan |fr - 6). 
2r sin £ (#--#) 

logsin|(£+P) =9.9475 
logcscl(^-JB) =0.7930 
log tan i (g - b) = 8.9023 

log tan ip = 9.6428 

Jp = 23°43.1' 

.-. p = 47° 26.2'. 

47° 26.2' = 2846.2'. 
360° = 21600'. 
Circumference of earth = 7912 x 3.1416. 

/. p (in miles) = ?§^ x 7912 X 3.1416. 

^luOO 

log 2846.2 = 3.4542 

colog 21600 = 5.6655 

log 7912 = 3.8983 

log 3.1416 = 0.4971 

log/> = 3.5151 
.-. p = 3274.3. 

Therefore the shortest distance between the places is 3274.3 miles ; 
the bearing of Greenwich from Boston is N. 53° 7.1' E., and of Boston 
from Greenwich, N. 71° 39.3' W. 



CHAPTER XII. — PAGE 134. 87 

2. Let C and V denote the positions of Calcutta and Valparaiso, 
respectively, and P the north-pole. 

Denote the arcs PV f PC, and CVby c, v, and;?. 
Then in the triangle PCV, 

Z P = 88° 19' + 71° 42' = 160° 1', c = 90° + 33° 2' = 123° 2', 
v = 90° -22° 33' = 67° 27'. 
By Art. 162, we have : 

tanKC-r^ rjfr-^ cotiP, 
sin J (c + i?) 

i + + 

tanJ(C+ F) = cos f ^"^ cotfrP. 

cos i (c + v) 

From the data, J(c - 1?)=27° 47.5', %(c + v)= 95° 14.5', J P= 80° 0.5'. 

logsinJ( c - v ) =9.6686 logcosJ( c — ") =9.9468 

log esc i (c + v) = 0.0018 log sec J ( c + v) = 1.0393 

log cot \ P =9.2459 log cot J P =9.2459 

log tan \ ( C - V) = 8.9163 log tan § ( <7 + V) = 0.2320 

... J (C - F) = 4° 42.9'. 180° - £ ( (7 + F) = 59° 37.5' 

.-. i(C+F)= 120° 22.5'. 

... a=j(o+F)+j(a-F)= 12505.4', 

and F= J (C + F) - § (C-F) = 115° 39.6'. 

. i sin i ( C + V) . , , v 
tan ^ = sinK(7lF) tanKc "^ 

log sin |(C+F) = 9.9359 

log esc J(C-F)= 1.0852 

log tan J (c — v) =9.7218 

log tan Jp =0.7429 

*^=79°45.2' 
.-. p= 159° 30.4'. 
159° 30.4' = 9570.4'. 
360° = 21600'. 
Circumference of earth = 7912 x 3.1416. 

.-. p (in miles) = ?5ZM x 7912 X 3.1416. 
" K J 21600 

log 9570.4 = 3.9809 

colog 21600 = 5.6655 

log 7912 =3.8983 

log 3.1416 = 0.4971 

logp = 4.0418 
,\ p = 11010. 



88 KEY TO ESSENTIALS OF TRIGONOMETRY. 

Therefore the shortest distance between the places is 11010 miles; 
the bearing of Calcutta from Valparaiso is N. 115° 39.6' E., or 
S. 64° 20.4' E., and of Valparaiso from Calcutta is N. 125° 5.4' W., 
or S. 54° 54.6' W. 

3. Let S and Q denote the positions of Sandy Hook and Queens- 
town, respectively, P the north pole, and Xthe intersection of the arc 
QS with the meridian of 50° W. 
Denote the arcs PQ and PS by s and q. 
Then in the triangle PQS, 

Z P = 74° 1' - 8° 19' = 65° 42', q = 90° - 40° 28' =49° 32', 
s = 90° -51° 50' = 38° 10'. 
By Art. 162, we have : 

tan l(Q-$) = s ! n J9* ~ s l cot i P, 

sin h (q + s) 

tan Kg +0) = CO8 * (9 ~ '> <**** 

cos i(q + s) 

Erom the data, 

J (g- a) = 5° 41', £(? + *) =43° 51', JP=32°51'. 
log sin i(q-s) = 8.9957 log cos J (q ■— i) = 9.9979 

log esc J [q + s) = 0.1594 log sec J (q + s) = 0.1419 

log cot J P =0.1900 log cot i P =0.1900 

logtanj(G-#) = 9.3451 logtanJ(Q+ #) = 0.3298 

... i(Q-^) = 12°28.9'. .-. J(Q+^)=64°55.5'. 

Then in the triangle PQX, 

Z P = 50° - 8° 19' = 41° 41', 

Z Q = i(Q + 8) + HQ - #) = 77° 24.4', PQ = 38° 10'. 
By Arts. 160 and 161, we have : 

tan J (PX- <jX) = g ? n ?;gTgpS tan 1 *«• 
sin J (Q + §PA) 

tan % (PX+ QX) = C0S i(Q-Q px ) tan J PQ. 
2 ^ ; cob HQ+QPX) * 

From the data, 

i(Q-QPX) = ll°51.V, i (Q + QPX) = 59° 32.7', JPiQ=19 5'. 

logsinJ(§- §P^) -9.4868 log cos |(§ - Q FX ) = 9 - 9785 

log esc i ( Q + QPX) - 0.0645 log sec l( Q + QPX) = 0.2951 

log tan iPQ = 9.5390 log tan } PQ = 9.5390 

log tan | (PX- §X) = 9.0903 log tan J (PX+ QX) = 9.8126 

.-. | (PX- §X) = 7° 1.2'. ... i (PX-f QX) = 33°0.4'. . 

.-. PX= i (PX+ QX) + * (PX- QX) = 40° 1.6'. 
Therefore the latitude of X= 90° - 40° 1.6' = 49° 58.4' N. 



CHAPTER XII. — PAGE 136. 89 

4. Denote the ares SP, SZ, and PZ (see Fig., p. 136) by z, p, and s, 
and their sum by 2 s'. 

Then in the triangle SPZ, 

z=90°- 18 c 30' = 71 c 24', p = 90° - 14° 18' = 75° 42', 
s = 90° -50° 13' = 39° 47'. 
By- Art. 158, 



coshP = J* [ns,sin ( s '-P^ 
\ sin s sin z 

From the data, 

s' = 93°26.5', s'— p = 17°4A.&. 

log sins' =9.9992 

logsin(s'-jj) = 9.4839 
log esc s =0.1939 

log esc z =0.0233 

2)9.7003 
log cos J P =9.8501 

/. JP=44 C 55', and P=89°50'. 

To 89° 50' corresponds 5 h. 59 m. 20 s. of time. 
Therefore the hour of the day is 6h. m. 40 s. a.m. 
The difference between the local and Greenwich time is 2 h. 59 m. 
20 s., which corresponds to 44° 50' of longitude. 
Hence the longitude of the vessel is 44° 50' W. 

5. Denote the arcs SP, SZ, and PZ (see Fig., p. 135) by z, p, and s. 
Then in the triangle SPZ, 

z = 90° + 12° = 102°, s = 90° - 37° 48' = 52° 12', ZP= 60°. 
By Art 162, we have : 

tan* (Z- 8) = sin K*-s) cot i Pf 
sin J [z + s) 

tan i(Z+S) = C05 h ( ~ ~ S ] cot | P. 
cos \ (z + S) 

From the data, \{z - s) = 24° 54', \{z + s) = 77°6', J P = 30°. 

log sin }(* — *) = 9 - 624 3 log cos £ (z - s) = 9.9577 

log esc \(z + s) = 0.0111 log sec | ( s + s) = 0.6512 

log cot \ P =0.2386 log cot \P =0.2386 

log tan \ (Z- 8) = 9.8740 log tan | (Z + S) = 0.8475 

... ;(Z->S)=36 C 48.1 ; . .-. |(Z+ 5) = 81° 64.8'. 



90 KEY TO ESSENTIALS OF TRIGONOMETRY. 

tan^ = sin ^g+^) tanKg _ s) . 
sin J (Z — S) 
log sin J (Z + S) =9.9957 
log esc %(Z-S) = 0.2225 
log tan J (z-s) = 9.6667 
logtanjjo =9.8849 

.-. i p = 37° 29.6', and /> = 74° 59.2'. 
Therefore the altitude of the sun is 90° - 74° 59.2', or 15° 0.8'. 

6. Denote the arcs SP t SZ, and PZ (see Fig., p. 135) by z, p, and 
s, and their sum by 2 s'. 

Then in the triangle SPZ, 
z = 90° + 3° = 93°, p = 90° - 25° 46' = 64° 14', s = 90° + 37°49' = 127°49'. 

By Art. 158, cos J P = J sins ' sin (»' -*»! 

\ sins sin 2 

From the data, s' = 142° 31.5', s'-p = 78° 17.5'. 

log sin s' = 9.7842 

log sin (s'-p) = 9.9908 

log esc s =0.1024 

log esc z = 0.0006 

2)9.8780 

log cos J P = 9.9390 

/. iP = 29° 40', and P = 59° 20'. 
To 59° 20' corresponds 3h. 57 m. 20 s. of time. 
Therefore the hour of the day is 8h. 2 m. 40 s. a.m. 

7. Denote the arcs SP, SZ, and PZ (see Fig., p. 135) by z, p, and s. 
Then in the triangle SPZ, 

z - 90° - 15° = 75°, j» = 90°, s = 90° - 42° 21' = 47° 39'. 

Therefore in the polar triangle of SPZ, we have 

Z = 180° - 75° = 105°, P' = 180° - 90° = 90°, 

S f = 180° - 47° 39' = 132° 21'. 

+ - - 

By Art. 144, cosp f = cot S 1 cot Z'. 

log cot S f = 9.9598 
log cot Z = 9.4281 
log cos jo' =9.3879 
... p' = 75°51.6'. 

Then in the triangle SPZ, we have 

P=180°-/>' = 104°8.4'. 
To 104° 8.4 corresponds 6h. 56 m. 33.6 s. of time. 
Therefore the hour of the day is 5h. 3 m. 26.4 s. a.m. 



USE OF THE TABLES. — PAGES 2, 4. 



91 



USE OF THE TABLES. 



Page 2. 



3. log 80 = 1.9031. 

4. log 6.3 = 0.7993. 

5. log 298 = 2.4742. 

6. log .902 = 9.9552 - 10. 

7. Mantissa of 772 = .8876 

.3x6= 2 

■\ log .7723 = 9.8878 -10 

8. Mantissa of 105 = .0212 

.6 X 41 = 25 



.-. log 1056 = 3.0237 

9. Mantissa of 329 = .5172 
.4 X 13 = 5 



.-. log 3.294 = 0.5177 

10. Mantissa of 520 = .7160 
.5x8= 4 



/. log .05205 = 8.7164 -10 

11. Mantissa of 200 = .3010 
.8 X 22 = 18 

/. log 20.08 = 1.3028 



12. Mantissa of 924 = .9657 

.61 X 4 = 2 

/. log 92461 = 4.9659 

13. Mantissa of 403 = .6053 

.22 X 11 = 2 



/. log .40322 = 9.6055 -10 

14. Mantissa of 717 = .8555 

.8x6= 5 

/. log .007178 =7.8560 -10 

15. Mantissa of 518 = .7143 

.09 x 9 = 1 

.-. log 5.1809 = 0.7144 

16. Mantissa of 103 = .0128 

.65 X 42 = 27 

.-. log 1036.5 = 3.0155 

17. Mantissa of 866 = .9375 

.76 x 5 = 4 

/. log .086676 = 8.9379 -10 

18. Mantissa of 115 = .0607 

.07 X 38 = 3 

/. log .11507 = 9.0610 -10 



Page 4. 



4. Number corresponding to 
1.8055 = 63.9. 



5. Number corresponding to 
9.4487 - 10 = .281. 



92 KEY TO ESSENTIALS OF TRIGONOMETRY. 

6. 0.2165 13. 9.3178-10 

.2148 = mantissa of 164 .3160 = mantissa of 207 



11= 6 * 

27 21 



.-. Number corresponding = 1.646 .-. Number corresponding = .2079 



7. 3.9487 

.9484 = mantissa of 888 



14. 1.6482 

.6474 = mantissa of 444 



10 



/. Number corresponding = 8886 . Numbep corresponding _ 44 48 

8. Number corresponding to 

2.7364 = 545. 15. 7.0450 - 10 

.0414 = mantissa of 110 



9. 8.1648 - 10 36 

.1644 = mantissa of 146 39 " 



10. 



A= 1 

29 


.-. Number corresponding=.001109 


ber corresponding = .01461 
7.5209 - 10 


16. 4.8016 

.8014 = mantissa of 633 


.5198 = mantissa of 331 


?= 29 


11 = 8 
13 


7 


.*. Number corresponding = 63329 



.*. Number corresponding =.003318 

11. 4.0095 17 - 8-H44-10 

.0086 = mantissa of 102 _1139 = mantissa of 130 



42 



_5_ 
21 34" 



.Number corresponding = 10221 



/. Number corresponding = 0.1301 



12. 0.9774 18. 2.7015 

.9773 = mantissa of 949 .7007 = mantissa of 502 



1= 2 §_ 

4 9" _ 

.*. Number corresponding =9.492 ,\ Number corresponding = 502.9 



USE OF THE TABLES. — PAGES 6, 7. 



93 





Page 


6. 


3. log tan 35° 10' = 9.8479 




7. log tan 82° 0' = 0.8522 


2.7x9= 24 




9.3 X 31 = 31 


.-. log tan 35° 19' = 9.8503 


/. log tan 82° 3' 20" = 0.8553 


4. log sin 61° 50' =9.9453 




8. log sin 55° 10' = 9.9142 


.6x8= 5 




.9x1.8= 2 


/. log sin 61° 58' =9.9458 


.-. log sin 55° 11.8' = 9.9144 


5. log cot 12° 30' = 0.6542 




9. log cos 30° 40' = 9.9346 


5.9 X 4 = 24 




.8 X 2.5 = 2 


.-. log cot 12° 34' = 0.6518 


/. log cos 30° 42.5' = 9.9344 


6. log cos 26° 50' = 9.9505 




10. log "cot 48° 0' = 9.9544 


.6X6= 4 




2.5 X 3f-§ = 9 


.-. log cos 26° 56' = 9.9501 


.-. log cot 48° 3' 43" = 9.9535 




Page 


i 7. 


3. 0.9164 




7. 9.2279 


0.9109 = log tan 83' 


D 0' 
5.2 


9.2251 = log cos 80° 20' 


55 _ 
10.5 ~ 


^-= 3.8 
7.3 


.*. Angle corresponding = 83' 


D 5.2' 


.*. Angle corresponding = 80° 16.2' 


4. 9.9238 




8. 9.4700 


9.9236 = log cos 33° 


0' 
2.5 


9.4669 = log cot 73° 40' 


2 _ 

.8 


-?! = 6.6 
4.7 


\ Angle corresponding = 32° 


57.5' 


.-. Angle corresponding = 73° 33.4' 


5. 9.8630 




9. 9.1891 


9.8629 = log sin 46° 50' 


9.1863 = log sin 8° 50' 


1 _ 
1.2 


0.8 


!= 


\ Angle corresponding = 46° 50.8' 


.*. Angle corresponding = 8° 53.5' 


6. 0.2154 




1-0. 0.0502 


0.2127 = log cot 31° 


30' 
9.6 


0.0481 = log tan 48° 10' 


27 _ 
2.8 


21= 8.4 
25 



, Angle corresponding = 31° 20.4' ,\ Angle corresponding = 48° 18.4' 



94 



KEY TO ESSENTIALS OF TRIGONOMETRY. 



log sin 65° 10' = 9.9579 
.5X2= 1 



log sin 65° 12' = 9.9580 
log esc 65° 12' = 0.0420 



log cos 80° 0' =9.2397 
7.3 X 7.3 = 53 



log cos 80° 7.3' =9.2344 
log sec 80° 7.3' =0.7656 



Page 8. 
4. 



9.5997 = log cos. 
9.5978 = log cos 66° 40' 

JL9 _ 

2.9 



6.6 



.\ Angle corresponding = 66° 33.4' 



9.8112: 

9.8111 : 
1.4' 



: log sin. 

: log sin 40° 20' 

0.7 



.*. Angle corresponding = 40° 20.7' 



Page 10. 



3. nat. sin 3° 30' 

.00029 x 2 

nat. sin 3° 32' 

mantissa of 616 
7X.3 

.-. log sin 3° 32' =8.7898 

4. nat. cos 88° 10' = .03199 

.000291 X 7 = .00204 

nat. cos 88° 17' = .02995 



= .06105 

= .00058 

= .06163 

= .7896 

= 2 



mantissa of 299 : 
14 X. 5: 



.4757 

7 



.-. log cos 88° 17' = 8.4764 



5. nat. tan 2° 20' = 
.000291 X 8.2 = 

nat. tan 2° 28.2': 

mantissa of 431 = 
10 X. 4: 

.-. log tan 2° 28.2': 



.04075 
: .00239 

: .04314 

: .6345 
4 

: 8.6349 



6. nat. tan 50' = .014545 
.000291 X 5J£ = .001523 

nat. tan 55' 14" =.016068 



mantissa of 160 = .2041 

27 X .68 = 18 



log tan 55' 14" = 8.2059 
.-. log cot 55' 14" =1.7941 

7. 7.8702 

.8698 = mantissa of 741 



Number corresponding = .007417 

.007417 = nat. sin. 
.005818 = nat. sin 20' 



.001599 
.0002909 



= 5.497'= 5' 29.8" 



•. Angle corresponding = 


= 25' 29.8" 


8. 8.6150 






.6149 r 


= mantissa of 412 


1 _ 
11" 




1 



Number corresponding = .04121 

.04121 = nat. cos. 
.04071 = nat. cos 87° 40' 



.00050 
.000291 



= 1.718': 



1'43.1" 



Angle corresp. = 87° 38' 16.9" 



USE OF THE TABLES.— PAGES 10, 11. 



95 



9. 8.2892 

.2878 = mantissa of 194 



14. 

22' 



64 



Number corresponding = .019464 

.019464 = nat. cot, 
.017455 = nat. cot 89° 0' 



.002009 



: 6.916' : 
.0002905 

.*. Angle corresp. : 



6' 55" 



> 53' 5" 



10. 8.2131 = cologarithm. 

.2122 = mantissa of 163 



26 



35 



Number corresponding = .016335 

.016335 

.014545 = nat. tan 50' 

.001790 = 6151/= _ 6 , 91 „ 

.000291 

56' 9.1" 
.\ Angle corresp. = 89° 3' 50.9" 



Page 11. 



3. nat. sin 17° 30' = .3007 


5. .7385 




.00028 X 3 = .0008 


.7373 = nat. sin 
.0012 _ 
.0019 
.*. Angle corresponding = 


47° 30' 


.-. nat. sin 17° 33' = .3015 


6.3 




= 47° 36.3' 


4. nat. cos 75° 40' = .2476 


6. .9280 




.00029 X 8.3 = .0024 


.9272 = nat. cos 

.0008 

.0011 


22° 0' 


•. nat. cos 75° 48.3' = .2452 


7.3 



\ Angle corresponding = 21° 52.7' 



